Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ and the non-archimedean analytic space $X^{an} = \operatorname{sp} \mathcal{A}$. The former has an etale theory from the 1960's, and the latter from V. Berkovich's '93 IHES paper. In particular for an abelian etale sheaf $\mathcal{L}$ I can analytify the sheaf to get a sheaf on the etale topology of $X^{an}$, which I'll call $\mathcal{L}^{an}$.

Are there any comparison results between $H^i(X_{\acute{e}t},\mathcal{L})$ and $H^i(X^{an}_{\acute{e}t},\mathcal{L}^{an})$?

The comparison theorems in that paper seem to work for algebras of finite type over $K$, or of finite type over an affinoid base. It doesn't seem directly possible to set it up so that the derived pushforward $R^q\varphi_*$ maps to $\operatorname{sp} K$.

I am principally interested in smooth affinoids over a discretely valued field, and $\mathcal{L}$ a locally constant sheaf of finite abelian groups whose orders are prime to the characteristic of the residue field.

I have a very convoluted argument in mind using Berkovich's most recent pre-print but I imagine there is an easier way.

EDIT: I suppose I should probably indicate that I've put some thought into this: You can take a closed immersion of $X^{an}$ into some ball (e.g. higher dimensional analogues of $E(0,r) \times D(0,s)$). This closed immersion is algebraically of finite type, and so we can use the comparison theorem to compare the two push-forwards. This plus the Leray spectral sequence would reduce the problem to showing the analogous result for constructible sheaves on balls. However there doesn't really seem to be any tools that I can obviously use to attack this.