In this talk we introduce the analytic de Rham stack for rigid varieties over Q_p (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.