$\mathcal{D}^{\infty}$-modules on smooth rigid analytic varieties and locally analytic representations

In this article, we construct the abelian category of coadmissible $p$-adic $\mathcal{D}^{\infty}$-modules on a smooth rigid analytic variety over a complete discrete valued field. We also consider equivariant $\mathcal{D}^{\infty}$-modules and prove a $p$-adic analogue of the Beilinson-Bernstein localization theorem for admissible locally analytic representations.