Yes, there is.
What happens to Satan?
The New Testament is the story of Christ's victory over sin and death. This happened with his death and resurrection, but the final accomplishment of his victory is delayed until the end of time. Matthew 25:31-46 quotes Jesus saying that at that time, he will return to judge all people.
1 Corinthians 15:22-26 explains (NIV):
For as in Adam all die, so in Christ all will be made alive. But each in turn: Christ, the firstfruits; then, when he comes, those who belong to him. Then the end will come, when he hands over the kingdom to God the Father after he has destroyed all dominion, authority and power. For he must reign until he has put all his enemies under his feet. The last enemy to be destroyed is death.
The book of Revelation covers this topic in detail. Towards the end (20:10) we read:
And the devil, who deceived them, was thrown into the lake of burning sulphur, where the beast and the false prophet had been thrown. They will be tormented day and night for ever and ever.
This is the final end of the devil (and of hell and death, 20:14, and "anyone whose name was not found written in the book of life", 20:15). God's choice for Satan is not annihilation, but eternal suffering, and that only begins once the "heavens and earth pass away" (21:1). Until then, Satan remains active in the world and retains the ability to tempt people.
Can God annihilate Satan, or anything else?
The glib answer is "Sure, why not?". Aquinas taught that God certainly can annihilate anything (Summa Theologica 104.3) on the grounds that God is omnipotent. But he then (104.4) quotes Ecclesiastes 3:14,
I know that everything God does will endure forever; nothing can be added to it and nothing taken from it.
to say that he chooses not to.
Why eternal torment, not just annihilation?
Augustine discusses this in City of God 21.11. His basic argument is that our sin (equally, the devil's) is deserving of infinite punishment because it is an offence against God, and therefore infinitely severe. Aquinas (Summa Theologica 99.1) expounds some other reasons as well:
For he who falls into mortal sin of his own will puts himself in a state whence he cannot be rescued, except God help him: wherefore from the very fact that he is willing to sin, he is willing to remain in sin for ever. [...] Thus if a man were to throw himself into a pit whence he could not get out without help, one might say that he wished to remain there for ever, whatever else he may have thought himself.
Another and a better answer is that from the very fact that he commits a mortal sin, he places his end in a creature; and since the whole of life is directed to its end, it follows that for this very reason he directs the whole of his life to that sin, and is willing to remain in sin forever, if he could do so with impunity.
Aquinas also says that since, once you have died, you no longer have the ability to choose, you are equally deserving of punishment or reward at any future point after your death. Some people do think that God's mercy will extend to forgiving all people, and even Satan himself, or that we will get another chance after death to accept God. Augustine doesn't have a very good argument against it - he just calls them "tender-hearted" and moves on. This area is highly contested even today.
There is a sense, then, in which being in hell forever is a fitting end for those who choose it (as Judas is called "the one who chose to be lost" or "the son of perdition" in John 17:12). Under the interpretation of hell as eternal separation from God, those who have chosen to be separated from God (by remaining in sin) get exactly what they wanted, except it turns out not be as nice as they had hoped.
This is pretty much exactly what Dante describes in the Divine Comedy. In his Hell, the punishment sinners receive isn't some ironically appropriate retribution - it is the sin itself, but without any comforting illusion. It's also what Lewis writes about in The Great Divorce, except that he imagines people being able to choose even after death. Lewis's grey city is just what its inhabitants want it to be, unfortunately.