  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.
