  We construct $p$-adic $L$-functions associated with $p$-refined cohomological
cuspidal Hilbert modular forms over any totally real field under a mild
hypothesis. Our construction is canonical, varies naturally in $p$-adic
families, and does not require any small slope or non-criticality assumptions
on the $p$-refinement. The main new ingredients are an adelic definition of a
canonical map from overconvergent cohomology to a space of locally analytic
distributions on the relevant Galois group and a smoothness theorem for certain
eigenvarieties at critically refined points.
