Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we can find a surjection $\mathcal{O}_K \left\langle X_1,\dots,X_n \right\rangle \twoheadrightarrow A^\circ$? The answer is "always" if $K$ is discretely valued, or $K$ is stable and the value group $|K^\ast|$ is divisible (in particular if $K$ is algebraically closed). This follows from some stuff in sections 6.3-6.4 of Bosch-Guntzer-Remmert, which in fact proves much more.

Surely the answer is not "always", but I'm having an annoyingly difficult time constructing a counterexample. I suspect that if $p>2$ and $K=\widehat{\mathbf{Q}_p(p^{1/p^\infty})}$, the power-bounded subring of $A= K \left\langle X,\frac{X^2}{p} \right\rangle$ is not topologically of finite type, but I wasn't quite able to prove it. Any help would be appreciated!