$\mathbb{N}$-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field $\mathbb{K}$ are studied. It is proven that the behaviour of these maps when $\mathbb{K}$ is not locally compact is very different from the case of real- or complex-valued functions: in general, for $\mathbb{N}$-compact spaces $X$ and $Y$, the existence of a (weakly) separating additive map $T: C^* (X)\rightarrow C^* (Y)$ implies that $X$ and $Y$ are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of $X$ and $Y$. Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.