Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? If so I might also ask if it is possible that an exact admissible sequence $X \to Y \to Z$ of bounded morphisms of $k$-Banach spaces leads to an exact admissible sequence $X \widehat{\otimes}K \to Y\widehat{\otimes}K \to Z\widehat{\otimes}K$ (or even conversely also)? Note: if we only demanded that $K$ is non-trivially valued then such a fact appears in Berkovich's book.



Spherical completions and flatness
Let me try to give an answer. In fact, I think that the statements you want always hold, i.e. for any complete valued field extension $K/k$. Which makes wonder whether I have missed something obvious... Anyway, writing things up clearly should help us start a discussion, so I will give it a try.


As regards the first question about the isometric embedding $X \to X\hat{\otimes}\_k K$, it can be deduced from lemma 3.1 (with $A=k$, $B=K$ and $C=X$) from my paper "Les espaces de Berkovich sont angéliques" in Bulletin de la SMF 141 (2), p. 267-297 (2013) (<http://smf4.emath.fr/Publications/Bulletin/141/html/smf_bull_141_267-297.php>). It eventually relies on arguments from BGR about the existence of $\alpha$-cartesian bases for finite-dimensional spaces over complete valued fields.


As I said in the comments above, the second question about the admissible exact sequence is dealt with by Gruson in "Théorie de Fredholm p-adique" Bulletin de la SMF 94, p. 67-95 (1966) (www.numdam.org/numdam-bin/fitem?id=BSMF\_1966\_*94*\_67\_0). 


