Abstract: We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra -- the Poisson analog of the standard notion of a restrictted Lie algebra -- and we prove a version of Darboux Theorem valid in positive characteristic setting.