Let $k$ be a nonArchimedean 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 nontrivially valued then such a fact appears in Berkovich's book.

$\begingroup$ Just to be sure: you do not assume that the field $k$ is complete, do you? $\endgroup$ – Jérôme Poineau Oct 7 '13 at 8:32

$\begingroup$ Thinking a little further, I do not see how you could use anything from Berkovich's book if your base field is not complete. Then your statements would be classical I believe. Could you clarify this point please? (And maybe also provide a more precise reference to Berkovich.) $\endgroup$ – Jérôme Poineau Oct 7 '13 at 8:42

$\begingroup$ Dear Jerome, Yes, my base field is complete. I forgot to mention that. $\endgroup$ – Oren BenBassat Oct 9 '13 at 18:26

$\begingroup$ I was talking about Proposition 2.1.2 from Berkovich's book. He shows that you can pass to a situation where the field is nontrivially valued. I am looking for a similar way to pass to the situation where the field is spherically complete. $\endgroup$ – Oren BenBassat Oct 9 '13 at 18:29

$\begingroup$ Dear Oren, OK, thanks. I have other questions then :) I have to say I am a little puzzled by what you asked, since I thought that what you want holds for arbitrary extensions $K/k$. For example, the second part should follow from Gruson's article "Théorie de Fredholm $p$adique" (numdam.org/numdambin/fitem?id=BSMF_1966__94__67_0). Am I missing something here? Do you have examples of extensions where you statements fail? $\endgroup$ – Jérôme Poineau Oct 10 '13 at 6:44
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. 267297 (2013) (http://smf4.emath.fr/Publications/Bulletin/141/html/smf_bull_141_267297.php). It eventually relies on arguments from BGR about the existence of $\alpha$cartesian bases for finitedimensional 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 padique" Bulletin de la SMF 94, p. 6795 (1966) (www.numdam.org/numdambin/fitem?id=BSMF_1966_94_67_0).

$\begingroup$ Dear Jerome, thanks very much. I still will probably want to speak about it with you more, but because it will be slow, I approved this answer in the meantime! $\endgroup$ – Oren BenBassat Oct 14 '13 at 16:18