Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$
be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) induces a commutative square of locally G-ringed spaces
$\require{AMScd}$
\begin{CD}
X^\mathrm{rig} @>f^\mathrm{rig}>> Y^\mathrm{rig}\\
@V V V= @VV V\\
X @>>f> Y
\end{CD}
Moreover let $\mathcal{E}$ be a coherent sheaf on $X$. Does then
$$ (R^qf_*(\mathcal{E}))^\mathrm{rig}\cong R^qf^{\mathrm{rig}}_*(\mathcal{E}^\mathrm{rig}) $$
hold in the category of coherent sheaves on $Y^{\mathrm{rig}}$?
This is at least known for the usual complex analytification over $\mathbb{C}$ shown by Serre. That's why I'm rather optimistic.
If it is easier/better to avoid Grothendieck topologies and work instead with locally ringed spaces then one can also replace rigid analytification with adic analytification. As the resulting topoi are equivalent the result should be the same.
Best Answer
Brian Conrad give a positive answer in http://www.numdam.org/item/10.5802/aif.2207.pdf Remark A.1.1 p. 1113/1114.