Yes. There's two close matches for what you're looking for.
The not-as-close-but-interesting one
These are called agrapha, or "not written". They aren't as exhaustive as the hadith, but they are the closest you can get within modern scholarship. They are often distinguished from the 114 sayings in the Gospel of Thomas (which now only attests to certain agrapha) as well as be attributed to Jesus by others, not simply put in his mouth (a la the Pistis Sophia or the Didache). Examples of such agrapha include:
"It is a more blessed thing to give, rather than to receive." — Acts 20:35 , likely circulated independently as a saying of Jesus in the first decade after his death
"He who is near me is near the fire; and he who is far from me is far from the kingdom [of God]." — Origen, Homilies on Jeremiah 20:3; independently attested to in Logion 82 of the Coptic Gospel of Thomas
"Be skilled money-changers, rejecting some things, but holding fast that which is good." — Clement's Stromata 1:28:177
"In whatever things I take you, in these I shall judge you." — Justin Martyr, Dialogue with Trypho, chapter 47, paragraph 4
"Those among you who desire to see me and inherit my kingdom must receive me through sorrows and toil." — The General Epistle of Barnabas 7:11
Clement of Rome (not to be confused with the Clement of Alexandria mentioned above) exhorted his audience in First Clement [chapters 2:1, 10:1, 13:1-3, and 46:8] to hear and obey the 'words of Jesus our Lord,' although based on these quotations we can assume that he was referring to an oral tradition that had been preserved by the Corinthian community, which may or may not make it agrapha.
What I think is pretty darn close
Furthermore, Papias of Hierapolis wrote or compiled a work entitled Exposition of the Sayings of Our Lord, which has been lost except for ten fragmentary quotations by Eusebius Pamphyli Theophanies 4:12 and Irenaeus of Gaul Against Heresies, 5:33:3-4. While we obviously can't be sure of its content based on these, it is more than likely that a few agrapha or otherwise unattested sayings of Jesus made their way into the text; which makes its loss tragic indeed for the New Testament scholastic community . One saying is found in Irenaeus' magnum opus, and can be found here:
"The days will come in which vines will grow, each having ten thousand shoots, and on each shoot ten thousand branches, and on each branch ten thousand twigs, and on each twig ten thousand clusters, and in each cluster ten thousand grapes, and each grape, when pressed, will give twenty-five measures of wine. And, when one of those holy ones takes hold of a cluster, another cluster will cry, 'I am the better cluster; take me, bless the Lord through me!'"
"In like manner," the Lord declared, "a grain of wheat also will generate ten thousand heads, and each head will have ten thousand grains, and each grain five pounds of clear and clean flour, worth twice their weight. And the remaining fruits and seeds and manner of fields will follow after in congruence with these, and all the animals using these foods which are taken from the earth will in turn become peaceful and consenting, subject to men in every command."
— Irenaeus of Lyon, Adversus Hæreses, Book V, ch. xxxiii, para. 3, 4
As for if there is any Christian equivalent to the Islamic hadith, I'd have to say that were Papias' Expositions still extant or preserved to a much larger degree, then it would take the cake. It provides quotations of Jesus, much like the hadith does of the prophet Muhammad and an explanation of what the author(s) believe he meant by this, probably chock-full of scriptural references. The deeds (miracles and short stories) of Jesus are in what scholars call the 'signs gospel' which is only hypothetical and was (if it ever existed) likely from the same community that wrote the gospel of John; in any case, Expositions of the Sayings of Our Lord comes the closest to your definition of the Muslim hadith.