PI Hopf algebras of prime characteristic

We study the conditions when a cocommutative Hopf algebra of prime characteristic has a polynomial identity (as an algebra). In the case of characteristic 0, this question reduces to smash products of group algebras and universal enveloping algebras, so the answer is known. In characteristic p, however, there are connected Hopf algebras other than restricted envelopes. We find necessary and sufficient conditions of being PI for some extensive classes of such algebras as well as their smash products with group algebras.