In the first part of this talk we are going to define certain integral structures, depending on a congruence level, for the sheaves of differential operators on a formal scheme which is a blow-up of a formal scheme which itself is formally smooth over a complete discrete valuation ring of mixed characteristic. When one takes the projective limit over all blow-ups, one obtains the sheaf of differential operators on the associated rigid space, introduced independently by K. Ardakov and S. Wadsley. In the second part we will explain what it means that formal models of flag varieties are D-affine (this concept is analogous to that of Beilinson-Bernstein and Brylinski-Kashiwara in the algebraic context). If time permits, we will explain an example which illustrates that methods and results from rigid cohomology can be used in connection with those sheaves to analyze locally analytic representations of p-adic groups. This is joint work with C. Huyghe, D. Patel, and T. Schmidt.