  In this article we study modules endowed with a ultrametric, from the point
of view of the geometric notion $C$-minimality. We give a complete
characterization of $C$-minimal valued modules over non-commutative rings of
skew polynomials of the form $R:=K[t;\varphi]$, where $K$ is a field, $\varphi$
an endomorphism of $K$ and $R$ is the $K$-algebra generated by $t$, such that
$at=ta^{\varphi}$ for $a\in K$. We deduce for instance that the ring of Puiseux
series over a finite field $\mathbb{F}$ of characteristic $p>0$, as a valued
module over $\mathbb{F}[t;x\mapsto x^p]$ is $C$-minimal. Moreover, any
ultraproduct $\mathcal{K}$, of algebraically closed valued fields
$\mathcal{K}_{p^n}$ of characteristic $p>0$, endowed each with the morphism
$x\mapsto x^{p^n}$, following a ultrafilter $U$ over $\{p^n\ |\ n\in
\mathbb{N}, \, \text{et} \; p \; \text{prime}\}$, equipped with the {\it
non-standart Frobenius}, i.e., the map $\sigma_{U}:=\lim_{U} x \mapsto
x^{p^n}$, is $C$-minimal as a $\mathcal{K}[t;\sigma]$-valued module.
