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In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field K are studied. It is proven that the behaviour of these maps when K is not locally compact is very different from the case of realor complex-valued functions: in general, for Ncompact spaces X and Y , the existence of a (weakly) separating additive map T : C∗(X)→ 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-Čech compactifications of X and Y . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

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