Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in V^\times$, one has $x^y:=yxy^{-1}\in F$. Can one find a finite number of elements $b_1,\dotsc,b_r\in F$ and an $e\in\mathbb N$, denoting $B_i:=\{x\in V\mid v(b_i-x)\ge e\}$, such that $B_i\cap B_j=\emptyset$ and $F\subset \bigcup_{i=1}^rB_i$ and for all $1\le i\le r$ and for all $y\in V^\times$, $yB_iy^{-1}\subset B_i$.

That's clearly true in the commutative case, but I do not if it is true when $K$ is not commutative.