I will describe a construction which associates a canonical $p$-adic L-function with a refined cohomological Hilbert modular form $(\pi, \alpha)$ under some mild and natural assumptions. The main novelty is that we do not impose any hypothesis of “small slope” or “noncriticality” on the allowable refinements. Over $\mathbb{Q}$, this result is due to Bellaiche. Our strategy for dealing with critical refinements is roughly parallel to his, and in particular relies on a careful study of the local geometry of eigenvarieties at classical (but possibly critical) points. This is joint work with John Bergdall.