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.