(Joint with Ananth Shankar and Ben Bakker) Given an (exceptional) Shimura variety S, we prove the existence of integral canonical models for S at all sufficiently large primes. Our method passes through finite characteristic and relies on a partial generalization of the work of Ogus-Vologodsky. As applications, we prove analogues of tate semisimplicity in finite characteristic, CM lifting theorems for ordinary points, and the Tate isogeny theorem for ordinary points.