Some of the modern vocabulary of the determinism/free will debate is not found in the writings of the Ante-Nicene Fathers, but within their writings there is clear & consistent support of the view that humans have free will, and will bear consequences for how they use it.
Irenaeus' Manifesto on Free Will
Irenaeus of Lyons was a huge champion of human free will, and wrote one of the most extensive defenses of it (this is a condensed version):
This expression [of our Lord], How often would I have gathered your children together, and you would not, set forth the ancient law of human liberty, because God made man a free [agent] from the beginning, possessing his own power, even as he does his own soul, to obey the behests of God voluntarily, and not by compulsion of God. For there is no coercion with God...therefore does He give good counsel to all. And in man...He has placed the power of choice...so that those who had yielded obedience might justly possess what is good, given indeed by God, but preserved by themselves. On the other hand, they who have not obeyed...Rejecting therefore the good, and as it were spuing it out, they shall all deservedly incur the just judgment of God, which also the Apostle Paul testifies in his Epistle to the Romans, where he says, But do you despise the riches of His goodness, and patience, and long-suffering, being ignorant that the goodness of God leads you to repentance? But according to your hardness and impenitent heart, you store to yourself wrath against the day of wrath, and the revelation of the righteous judgment of God. But glory and honour, he says, to every one that does good. God therefore has given that which is good, as the apostle tells us in this Epistle, and they who work it shall receive glory and honour, because they have done that which is good when they had it in their power not to do it; but those who do it not shall receive the just judgment of God, because they did not work good when they had it in their power so to do.
But if some had been made by nature bad, and others good, these latter would not be deserving of praise for being good, for such were they created; nor would the former be reprehensible, for thus they were made [originally]. But since all men are of the same nature, able both to hold fast and to do what is good; and, on the other hand, having also the power to cast it from them and not to do it — some do justly receive praise...but the others are blamed, and receive a just condemnation, because of their rejection of what is fair and good. And therefore the prophets used to exhort men to what was good, to act justly and to work righteousness, as I have so largely demonstrated, because it is in our power so to do, and because by excessive negligence we might become forgetful, and thus stand in need of that good counsel which the good God has given us to know by means of the prophets.
For this reason the Lord also said, Let your light so shine before men, that they may see your good deeds, and glorify your Father who is in heaven...And again, The servant who knows his Lord's will, and does it not, shall be beaten with many stripes. And, Why call me, Lord, Lord, and do not the things which I say?...All such passages demonstrate the independent will of man, and at the same time the counsel which God conveys to him, by which He exhorts us to submit ourselves to Him, and seeks to turn us away from [the sin of] unbelief against Him, without, however, in any way coercing us.
No doubt, if any one is unwilling to follow the Gospel itself, it is in his power [to reject it], but it is not expedient. For it is in man's power to disobey God, and to forfeit what is good...we should not use our liberty as a cloak of maliciousness, for this is not expedient. ...If then it were not in our power to do or not to do these things, what reason had the apostle, and much more the Lord Himself, to give us counsel to do some things, and to abstain from others? But because man is possessed of free will from the beginning, and God is possessed of free will, in whose likeness man was created, advice is always given to him to keep fast the good, which thing is done by means of obedience to God.
And not merely in works, but also in faith, has God preserved the will of man free and under his own control...man is in his own power with respect to faith. And for this reason, he that believes in Him has eternal life while he who believes not the Son has not eternal life, but the wrath of God shall remain upon him. In the same manner therefore the Lord, both showing His own goodness, and indicating that man is in his own free will and his own power, said to Jerusalem, How often have I wished to gather your children together, as a hen [gathers] her chickens under her wings, and you would not! Wherefore your house shall be left unto you desolate.
...[men] were made rational beings, endowed with the power of examining and judging, and were not [formed] as things irrational or of a [merely] animal nature, which can do nothing of their own will, but are drawn by necessity and compulsion to what is good, in which things there is one mind and one usage, working mechanically in one groove, who are incapable of being anything else except just what they had been created. But upon this supposition, neither would what is good be grateful to them, nor communion with God be precious, nor would the good be very much to be sought after, which would present itself without their own proper endeavour, care, or study, but would be implanted of its own accord and without their concern. Thus it would come to pass, that their being good would be of no consequence, because they were so by nature rather than by will, and are possessors of good spontaneously, not by choice; and for this reason they would not understand this fact, that good is a comely thing, nor would they take pleasure in it. For how can those who are ignorant of good enjoy it? Or what credit is it to those who have not aimed at it? And what crown is it to those who have not followed in pursuit of it, like those victorious in the contest? (Against Heresies 4.37.1-6)
Clement of Rome (late 1st century)
Clement believed we had to submit our will to God's:
therefore, that we be prompt in the practice of well-doing; for of Him are all things. And thus He forewarns us: "Behold, the Lord [cometh], and His reward is before His face, to render to every man according to his work." He exhorts us, therefore, with our whole heart to attend to this, that we be not lazy or slothful in any good work. Let our boasting and our confidence be in Him. Let us submit ourselves to His will. (1 Clement 34:2-4)
Note all the decisive action-verbs on the part of the believers:
35:5 And how will this be, beloved? If our mind be established by faith toward God; if we seek out what is pleasant and acceptable in his sight; if we perform such things as harmonize with his blameless will, and follow in the way of truth, casting from us all unrighteousness and lawlessness... (1 Clement 35:5)
The Shepherd of Hermas (late 1st or early 2nd century)
Hermas believed in the conditional nature of forgiveness, and that our own choices impact our outcome & guilt. We cannot earn the remedy, but it is given conditionally, upon repentance (and perseverance).
3 "I too, Sir," I say, "declare to every man the mighty works of the
Lord; for I hope that all who have sinned in the past, if they hear
these things, will gladly repent and recover life."
4 "Continue therefore," said he, "in this ministry, and complete it
unto the end. For whosoever fulfill his commandments shall have life;
yea such a man (shall have) great honor with the Lord. But whosoever
keep not his commandments, fly from their life, and oppose him, and
follow not his commandments, but deliver themselves over to death; and
each one becometh guilty of his own blood. But I bid thee obey these
commandments, and thou shalt have a remedy for thy sins. (Sim. 10
Epistle of Barnabas (late 1st or early 2nd century)
The author of Barnabas believes people can choose to reject salvation:
For he who
doeth these things shall be glorified in the Kingdom
of God, but he who chooseth the contrary things shall
perish together with his works. On this account is the
resurrection; on this account is the retribution. (Barnabas 21:1)
*note that previously Barnabas has acknowledged that this is true of people who already know God's ordinances.
Ignatius of Antioch (early 2nd century)
Ignatius believed our end state would be the result of our choices.
the choice of two things, death and life, is placed at once before us (Magnesians 5:1)
God the Father through Jesus Christ, through whom unless we attain voluntarily to die unto his passion, his life is not in us. (Magnesians 5:2)
Polycarp of Smyrna (early 2nd century)
Polycarp recognized the conditional non-deterministic, nature of salvation:
Now He that raised Him from the dead will raise us also; if we
do His will and walk in His commandments and love the things which He
loved (Epistle to the Philippians 2:2)
Justin Martyr (mid 2nd century)
Justin believed God created men free to reason & to choose, and that accordingly, these free & reasoning creatures could be held accountable for their actions:
God, wishing men and angels to follow His will, resolved to create them free to do righteousness; possessing reason, that they may know by whom they are created, and through whom they, not existing formerly, do now exist; and with a law that they should be judged by Him, if they do anything contrary to right reason: and of ourselves we, men and angels, shall be convicted of having acted sinfully, unless we repent beforehand. But if the word of God foretells that some angels and men shall be certainly punished, it did so because it foreknew that they would be unchangeably [wicked], but not because God had created them so. (Dialogue with Trypho ch. 141)
Justin acknowledges conscious actions involved in repentance, which are prerequisite to forgiveness of sins:
having repented of his sins, that he may receive remission of them from God; and not as you deceive yourselves, and some others who resemble you in this, who say, that even though they be sinners, but know God, the Lord will not impute sin to them. We have as proof of this the one fall of David, which happened through his boasting, which was forgiven then when he so mourned and wept, as it is written. But if even to such a man no remission was granted before repentance, and only when this great king, and anointed one, and prophet, mourned and conducted himself so, how can the impure and utterly abandoned, if they weep not, and mourn not, and repent not, entertain the hope that the Lord will not impute to them sin? (ibid ch. 141)
More Irenaues of Lyons (late 2nd century)
to believe in Him is to do His will (Against Heresies 4.6.5)
Irenaeus scathingly rebuked those who:
hold that they shall be entirely and undoubtedly saved, not by means of conduct, but because they are spiritual by nature. (ibid 1.6.2)
Shoots of determinism were seen among the Gnostics in Irenaues' day (see more below), and he rejected this view. He also spoke of the reward:
which does not encircle us of its own accord (ibid 4.37.7)
Irenaeus regularly speaks of human choice, and that God does not compel us to do/be good:
[man] may with judgment make choice of the better things; and that he may never become indolent or neglectful of God's command; and learning by experience that it is an evil thing which deprives him of life, that is, disobedience to God, may never attempt it at all, but that, knowing that what preserves his life, namely, obedience to God, is good, he may diligently keep it with all earnestness. Wherefore he has also had a twofold experience, possessing knowledge of both kinds, that with discipline he may make choice of the better things. (ibid 4.39.1)
but the man who does not obtain it is the cause to himself of his own imperfection. Nor, [in like manner], does the light fail because of those who have blinded themselves; but while it remains the same as ever, those who are [thus] blinded are involved in darkness through their own fault. The light does never enslave any one by necessity; nor, again, does God exercise compulsion upon any one unwilling to accept the exercise of His skill. Those persons, therefore, who have apostatized from the light given by the Father, and transgressed the law of liberty, have done so through their own fault, since they have been created free agents, and possessed of power over themselves. (ibid 4.39.3)
Clement of Alexandria (late 2nd century)
Clement believed people could turn away from the path of salvation:
wealth is of itself sufficient to puff up and corrupt the souls of its possessors, and to turn them from the path by which salvation is to be attained (Who is the Rich Man That Shall Be Saved? ch. 1)
Clement recognized the vital interplay of works with the gift of grace:
For it appears to me to be far kinder...to aid them in working out their salvation in every possible way; asking this of God, who surely and sweetly bestows such things on His own children; and thus by the grace of the Saviour healing their souls, enlightening them and leading them to the attainment of the truth; and whosoever obtains this and distinguishes himself in good works shall gain the prize of everlasting life. (ibid ch. 1)
And Clement acknowledged that knowledge without action/preparation results in falling short of what might have been:
But others rightly and adequately comprehend this, but attaching slight importance to the works which tend to salvation, do not make the requisite preparation for attaining to the objects of their hope. (ibid ch. 2)
Hippolytus of Rome (early 3rd century)
Hippolytus explains justification & judgement based what people have chosen to do:
Righteous is Your judgment. Of which voice the justification will be seen in the awarding to each that which is just; since to those who have done well shall be assigned righteously eternal bliss, and to the lovers of iniquity shall be given eternal punishment. (Against Plato ch. 3)
Cyprian of Carthage (3rd century)
After illustrating the point through the parable of the wise man and the foolish man, Cyprian observes:
But how can a man say that he believes in Christ, who does not do what Christ commanded him to do? Or whence shall he attain to the reward of faith, who will not keep the faith of the commandment? He must of necessity waver and wander, and, caught away by a spirit of error, like dust which is shaken by the wind, be blown about; and he will make no advance in his walk towards salvation, because he does not keep the truth of the way of salvation. (Treatise 1 - On the Unity of the Church ch. 2)
Cyprian saw the way of salvation as a walk, a journey, undoubtedly a journey aided, supported, and made possible by Christ, but a journey believers had to choose to start on and stay on.
Were there any exceptions?
Yes, there was one group of early Christians who believed in determinism: the Gnostics. Or, more appropriately, just a portion of the Gnostics. Gnosticism was disavowed by the Apostle John and generation after generation of Patristic writers who succeeded him (especially by Irenaeus!)(see discussion in Bercot Will the Real Heretics Please Stand Up p.66).
Centuries later Augustine, in his debate against the Pelagian heresy, would swing the pendulum far in the other direction, teaching a theology of grace that is not found in Patristic writings prior to the late 4th century, nor is it consistent with the cultural context of grace ("charis") of the New Testament.
In Paul's day, grace did not describe a "no-strings-attached, freebie". One could not receive a gift of charis and then just walk away, no, charis involved an agreement (a covenant or a treaty) and there were obligations of the recipient.
Charis has been described an an "asymmetric, reciprocal, gift relationship". A stronger party gave to a weaker party what the weaker party could never earn on their own, and as a result, expected loyalty/other things of the recipient. When Paul speaks of grace, he speaks of a two-way covenant, not a present that one can take and run.
Charis is not a loan--it can't be paid back--but the principle of reciprocity is inherent in the concept of grace--those who would remove reciprocity put an idea in Paul's mind that was never present in the Hellenistic world in which he lived & wrote.
(the previous 3 paragraphs are largely taken (and condensed) from the work of Greek scholar Brent Schmidt--a 10-page version of his thoughts can be found here; a much more lengthy version can be found here).
Did the Early Church (ante-Nicene period) believe in libertarian free will?
They did not call it this, but yes, they believed in a form of free will that we today would call "libertarian free will", and that a human's fate would depend upon the use of this free will.
Did the Early Church (ante-Nicene period) believe in determinism?
No, except for a fraction of the Gnostics.
Did the Early Church (ante-Nicene period) believe that salvation requires a synergistic cooperation between a free human being and God's grace, and that a person can freely choose to resist God's grace and lose their salvation?
The early church does not demonstrate an interest in the faith/works debate as we see it today. They believed in salvation by grace & they believed salvation did not come without obedience, and they did not see these as contradictory views.
The writings of the Ante-Nicene Fathers on grace & free will tend to be very consistent with the covenantal understanding of charis (grace) in the Hellenistic world; this historical-cultural understanding of grace would largely disappear from Christian writings in subsequent centuries.