$p$-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.