In this talk, we introduce the analytic de Rham stack for rigid varieties over Qp  (and more general analytic stacks). This object is an analytic incarnation of the (algebraic) de Rham stack of Simpson, and encodes a theory of analytic D-modules extending the theory of D-cap-modules of Ardakov and Wadsley. We mention how a very general six functor formalism can be construct in this set up, as well as other features such as Kashiwara equivalence and Poincaré duality for smooth maps.