Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be the a $k$-affinoid Tate algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is it true that there exists an element $g \in GL_n(A)$ such that $gLg^{-1} \subset GL_n(A^0)$?

This is true in the (well known) case of $A$ a finite extension of $\mathbb{Q}_p$. I tried to reproduce the same argument in the general case but I ran into problems at some point. Basically, we want to consider the submodule $T = \sum_{g \in L/G} g \cdot (A^0)^n$, where $G$ is a suitable open subgroup of $L$. In the case of a local field this is easily seen to be an $A^0$-lattice but I am not sure that it also holds for a general affinoid.

Any suggestions will be of great interest to me.

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be a $k$-affinoid algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is it true that there exists an element $g \in GL_n(A)$ such that $gLg^{-1} \subset GL_n(A^0)$?

This is true in the (well known) case of $A$ a finite extension of $\mathbb{Q}_p$. I tried to reproduce the same argument in the general case but I ran into problems at some point. Basically, we want to consider the submodule $T = \sum_{g \in L/G} g \cdot (A^0)^n$, where $G$ is a suitable open subgroup of $L$. In the case of a local field this is easily seen to be an $A^0$-lattice but I am not sure that it also holds for a general affinoid.

Any suggestions will be of great interest to me.