The de Rham stack for rigid analytic varieties

The de Rham stack of a smooth scheme of finite type over a field of characteristic zero is a geometric object that encodes the theory of $D$-modules in its theory of quasi-coherent sheaves. In this talk, using analytic geometry of Clausen and Scholze, I will explain how to construct similar de Rham stacks for rigid spaces (and some relatives) that encode the theory of solid $\hat{D}$-modules and locally analytic representations of $p$-adic Lie groups.