Any compact subset of $\bar{\mathbb{Q}}_{p}$ is in a finite extension of $\mathbb{Q}_p$? It is well-known that any compact subgroup of  $\bar{\mathbb{Q}}_{p}$ (algebraic closure of $\mathbb{Q}_p$) is in a finite extension of $\mathbb{Q}_p$. (Usually, its proof requires Baire category theorem.)
My question: does this hold for any compact set in $\bar{\mathbb{Q}}_{p}$?
i.e., Any compact subset of $\bar{\mathbb{Q}}_{p}$ is in a finite extension of $\mathbb{Q}_p$? 
It seems that my question is equivalent to the following: 'any compact subset of $GL_n(\bar{\mathbb{Q}}_p)$ is in $GL_n(E)$ where $E/\mathbb{Q}_p$ is a finite extension.'

Counterexample: $C:=\{p^n p^{1/n}\} \cup \{0\}$.
More generally, take the union of $\{0\}$ and (the set of the members of) any sequence $\alpha_n$ such that the corresponding sequence of values $(| \alpha_n |_p)_n$ has $0$ as its only accumulation point. There's plenty such sequences which are not contained in any finite extension of $\mathbb Q_p$.
