For a large class of smooth dagger spaces–rigid spaces with overconvergent structure sheaf–we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot's rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction.
"Finiteness of de Rham cohomology in rigid analysis." Duke Math. J. 113 (1) 57 - 91, 15 May 2002. https://doi.org/10.1215/S0012-7094-02-11312-X