Specific question on Banach space over nonarchimedean field

Question

Let K be an nonarchimedean field. We let $Ban(K)$ denote the category of $K$-Banach space with continuous linear maps and let $C$ be the category of normed K-Banach spaces ($V, ||$ $ || $) satisfying $||V||$ is contained in $|K|$ with norm decreasing linear maps. Then how to show that $Hom_{Ban(K)}(V,W)=Hom_{C}(V,W)\otimes \Bbb{Q}$ for two Banach space $V$, $W$. The statement for this fact is here at page $9$.

Answer 1

Given a continuous morphism, the preimage of the unit ball in the target contains a ball in the source. If we scale the given morphism by some scalar of sufficiently small absolute value (which we can assume lies in $\mathbb Q$, if $K$ has char. zero but positive char. residue field, which seems to be implicit in the question), it will take the unit ball in the domain into the unit ball in the target.

This shows that the inclusion $$Hom_C(V,W) \hookrightarrow Hom_{Ban(K)}(V,W)$$ induces a surjection $$Hom_C(V,W) \otimes \mathbb Q \to Hom_{Ban(K)}(V,W).$$ It is easy to see that this is in fact an isomorphism.