group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Let $X = Spec_{an}(A)$ be an affinoid space with affinoid algebra $A$.
Then for every finite cover of $X$ by affinoid domains, the corresponding Cech cohomology with coefficients in the structure sheaf $A$, or with coefficients in any finite Banach module over $A$, is concentrated in degree 0.
S. Bosch, U. Güntzer, Reinhold Remmert, section 8.2 of Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)
Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)
Doosung Park, Affinoid domains, lecture notes (pdf)