Let $K$ be a non-trivially non-archimedean valued field that is
complete with respect to the valuation $\left| \, \, \right| : K
\longrightarrow[0,\infty)$, let $X$ be a non-empty subset of $K$ without
isolated points. For $n \in \{ 0,1, \ldots \}$ the $K$-Banach
space $BC^n(X)$, consisting of all $C^n$-functions $X \longrightarrow K$ whose
difference quotients up to order $n$ are bounded, is defined in a
natural way. It is proved that $BC^n(X)$ is of countable type if
and only if $X$ is compact. In addition we will show that
$BC^{\infty}(X) : = \bigcap _n BC^n(X)$, which is a Fréchet space
with its usual projective topology, is of countable type if and
only if $X$ is precompact.