Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a field $K$ which is a finite extension of $\mathbb{Q}_p$. Then since $V$ is complete which implies that $f(V)$ is complete as $f$ is continuous. Now complete subspace of Hausdorff space is closed. That implies that $f(V)$ is closed. Now take any closed subset $L$ of $V$. Then $L$ is complete and using the same argument $f(L)$ is closed. So $f$ is a closed map. Is my proof correct?
If not the what are the condition in which $f$ can be a closed map?
There is a direct analog of closed Graph theorem of non-archimedean fields. see page 61, theorem 3.5 in Non-archimedean functional analysis. A. C. M. van Rooij.