In the 80s Goodwillie proved that the periodic cyclic homology in characteristic zero is rigid with respect to nilpotent extensions. In the early nineties Getzler constructed a Gauss-Manin connection on the periodic cyclic homology of a family of algebras. We extend these results to algebras over ${\mathbb F}_p.$ Namely, given such an algebra $A_0,$ we construct a version of its periodic cyclic complex over the $p$-adics, prove the invariance properties, and carry out an analogous construction for a Gauss-Manin superconnection for a family of algebras. This work is closely related to recent works of Petrov and Vologodsky, as well as Kaledin.