  The main purpose of this paper is to study restricted formal deformations of
restricted Lie-Rinehart algebras in positive characteristic $p$. For $p>2$, we
discuss the deformation theory and show that deformations are controlled by the
restricted cohomology introduced by Evans and Fuchs. Furthermore, for $p=2$, we
introduce a new cohomology complex and show that it fits with the deformation
theory of restricted Lie-Rinehart algebras in characteristic 2. In particular,
we study the structure and cohomology of restricted Heisenberg algebras.
