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We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathrm{c}ard\left(D\right)=\mathrm{d}ens\left(E\right)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathrm{d}ens\left(E\right)=\mathrm{d}ens\left(F\right)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $𝕂$ is homeomorphic to the non-archimedean Fréchet space ${𝕂}^{\mathbb{N}}$.

Let $(E,t)$ be a Hausdorff locally convex space. Either $(E,\sigma (E,E^{\text{'}}))$ or $(E^{\text{'}},\sigma (E^{\text{'}},E))$ is a $DF$-space iff $E$ is of finite dimension (THEOREM). This is the most general solution of the problem studied by Iyahen [2] and Radenovič [3].

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...