p-adic rigidity of eigenforms of infinite slope

We give a notion of p-adic families of Hecke eigenforms that allows for the slope of the forms be infinite at p. We prove that, contrary to the case of finite slope when every eigenform lives in a Hida or Coleman family, the only families of infinite slope are either twists of Hida or Coleman families with Dirichlet characters of p-power conductor, or non-ordinary families with complex multiplication. Our proof goes via a local study of deformations of potentially trianguline Galois representations, relying on work of Berger and Chenevier, and a global input coming from an analogue of a result of Balasubramanyam, Ghate and Vatsal on a Greenberg-type conjecture for families of Hilbert modular forms.