Let X be a smooth rigid analytic variety over a p-adic field. Ardakov-Wadsley introduced a completed sheaf of differential operators D-cap on X in order to adapt methods from geometric representation theory to the study of p-adic locally analytic representations. By working in the derived category of complete bornological D-cap-modules, we develop a theory of six operations and propose a definition of holonomic D-cap-modules, on which these operations are expected to give rise to a six-functor formalism (with some caveats). We then prove adjunction and base change results as well as stability of holonomicity under five of the six operations.