  We study the Banach algebras ${\rm C}(X, R)$ of continuous functions from a
compact Hausdorff topological space $X$ to a Banach ring $R$ whose topology is
discrete. We prove that the Berkovich spectrum of ${\rm C}(X, R)$ is
homeomorphic to $\zeta(X) \times {\mathcal M}(R)$, where $\zeta(X)$ is the
Banaschewski compactification of $X$ and ${\mathcal M}(R)$ is the Berkovich
spectrum of $R$. We study how the topology of the spectrum of ${\rm C}(X, R)$
is related to the notion of homotopy Zariski open embedding used in derived
geometry. We find that the topology of $\zeta(X)$ can be easily reconstructed
from the homotopy Zariski topology associated to ${\rm C}(X, R)$. We also prove
some results about the existence of Schauder bases on ${\rm C}(X, R)$ and a
generalisation of the Stone--Weierstrass Theorem, under suitable hypotheses on
$X$ and $R$.
