Let $k$ be a field with a non-archimedean complete valuation $|\ |$, $X$ a reduced rigid analytic space over $k$. The presheaf $\mathcal{O}^0$ which to an affinoid $U$ of $X$ attaches the ring $\mathcal{O}(U)^0$ of power-bounded elements of $\mathcal{O}(U)$ is a sheaf for the $G$-topology (isn't it?) for the condition for an $f \in \mathcal{O}(U)$ of being power bounded is equivalent to $|f(x)| \leq 1$ for all $x \in U$, a clearly local condition.

Is there a reference for this statement, and more generally a theory of "coherent" $\mathcal{O}^0$-modules, analog to Kiehl's theory of coherent $\mathcal{O}$-modules?