The most interesting answer I know to this question is the recent work of Albert Schwarz with Vadim Vologodsky, Ilya Shaprio (and Kontsevich), in which for example they use properties of the Frobenius action on p-adic cohomology to establish properties of the mirror map, see e.g. here, here, or here. In another interesting direction there are the papers of Candelas, de la Ossa and Rodriguez Villegas on Calabi-Yau manifolds over finite fields and "Dwork theory for physicists" here, here, and here.

The most interesting answer I know to this question is the recent work of Albert Schwarz with Vadim Vologodsky, Ilya Shaprio and Maxim Kontsevich, in which for example they use properties of the Frobenius action on p-adic cohomology to establish properties of the mirror map, see e.g. here, here, or here. In another interesting direction there are the papers of Philip Candelas, Xenia de la Ossa and Fernando Rodriguez Villegas on Calabi-Yau manifolds over finite fields and "Dwork theory for physicists" here, here, and here.