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We introduce a sheaf of infinite order differential operators (D) over cap on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Frechet-Stein algebras, and use this to define co-admissible sheaves of (D) over cap -modules. We prove analogues of Cartan's Theorems A and B for co-admissible (D) over cap -modules.