Let $K$ be an infinite algebraic extension of $\mathbf{Q}_p$ (not the algebraic closure). Let $L$ be the algebraic closure of the completion $\widehat{K}$.
Let $\mathbf{C}_p$ be the completion of the algebraic closure $\overline{\mathbf{Q}}_p$.
We have a map $\mathbf{C}_p \longrightarrow \widehat{L}$. Is this map always an isomorphism?