No! The correct expression is between you and me, NOT between you and I. When a pronoun is the object of a verb or preposition rather than the subject of a clause, you must use the pronoun’s objective case rather than its nominative case: me (not I); him (not he); her (not she); us (not we); them (not they).
The proliferation of between you and I is an example of what Bryan Garner (author of Garner’s Modern American Usage) aptly calls “hypercorrection.” This is how he explains the phenomenon:
Some people learn a thing or two about pronoun cases, but little more. They learn, for example, that it is incorrect to say It is me or Me and Jane are going to school now. . . . But this knowledge puts them on tenterhooks: through the logical fallacy known as “hasty generalization,” they come to fear that something is amiss with the word me — that perhaps it’s safer to stick to I. They therefore start using I even when the objective case is called for: “She had the biggest surprise for Blair and I [read me].” / “Please won’t you keep this between you and I [read me].” These are gross linguistic gaffes, but it is perennially surprising how many otherwise educated speakers commit them.