Can we say "Peter is sitting between you and I?" in formal writing?

"you and me" is correct in all aspects - formal and informal speech. "I" is always wrong.

If you ever wonder which is correct, say the sentence but leave out the other object (i.e. in this case "you"). The sentence would then be "Peter is sitting between me". Obviously that isn't a proper sentence, but it's a "true test" to determine the use of "me" or "I". "I" is a nomitive pronoun - "me" is an objective pronoun. "I am going to play basketball". You would know right away that "Me is going to play basketball" is incorrect.

So whenever you have a sentence that uses two objects (i.e. "you" and "I" in a comparative sense) just leave out the other object and listen to yourself say the sentence. You'll hear immediately if "I" is the wrong choice.

Peter is sitting between you and me. I is wrong because I is a subject not the object of a preposition. Between you and me is a prepositional phrase.

"Peter is sitting between you and me" sounds more formal.