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We study Banach spaces over a non-spherically complete non-Archimedean valued field K. We prove that a non-Archimedean Banach space over K which contains a linearly homeomorphic copy of $l^{\infty }$ (hence $l^{\infty }$ itself) is not a K-space. We discuss the three-space problem for a few properties of non-Archimedean Banach spaces.

We prove a non-archimedean Dugundji extension theorem for the spaces $C^{*}(X,𝕂)$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $𝕂$. Assuming that $𝕂$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T:C^{*}(Y,𝕂)\to C^{*}(X,𝕂)$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $𝕂$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular...