  Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers
and $\O$ be its integral ring. The convergent power series with coefficients in
$\O$ are studied as dynamical systems on $\O$. A minimal decomposition theorem
for such a dynamical system is obtained. It is proved that there are
uncountably many minimal subsystems, provided that there is a minimal set
consisting of infinitely many points. In particular, the complete detailed
minimal decompositions of all affine systems are derived.
