Does "an eye for an eye, a tooth for a tooth" appear in the Bible?

Yes, Matthew 5:38

Love Your Enemies
38 "You have heard that it was said, 'An eye for an eye, and a tooth for a tooth. 39 But I tell you, do not resist an evildoer. If someone strikes you on your right cheek, turn to him the other also. 40 If anyone wants to sue you to take away your shirt, give him your coat too. 41 Whoever compels you to go one mile, go with him two. 42 Give to the one who asks you, and do not turn away from the one who wants to borrow from you.

https://www.biblegateway.com/passage/?search=Matthew%205&version=EHV

To put it into context, the original quote comes from the Code of Hammurabi, a Babylonian law code from about 1800 BCE (so it was definitely old news by the time the Gospels were being written in the first century CE). It established the principle that punishment should be commensurate with the crime.

https://exclusivepapers.com/essays/ethics/an-eye-for-an-eye-a-tooth-for-a-tooth.php#:~:text=%E2%80%98An%20eye%20for%20an%20eye%2C%20a%20tooth%20for,punishment%20methods%20depending%20on%20the%20type%20of%20crime.

The sentiment is also clear in the Old Testament. Exodus 21, 23-25 (KJV) reads:

23 And if any mischief follow, then thou shalt give life for life,

24 Eye for eye, tooth for tooth, hand for hand, foot for foot,

25 Burning for burning, wound for wound, stripe for stripe.

https://www.biblegateway.com/passage/?search=exodus%2021&version=KJV

The message in Matthew is that one should be more forgiving and generous than that.