The analytic de Rham stack in rigid geometry

Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. We use this formalism to define the analytic de Rham stack in rigid geometry, extending the theory of $D$-cap-modules of Ardakov and Wadsley to the theory of analytic $D$-modules. We prove some foundational results such as the existence of a six functor formalism and Poincar\'e duality for analytic $D$-modules, generalizing previous work of Bode. Finally, we relate the theory of analytic $D$-modules to previous work of the author with Rodrigues Jacinto on solid locally analytic representations of $p$-adic Lie groups.