  The main objective of this article is to develop the theory of deformation of
$C^*$-algebras endowed with a group action, from the perspective of non-formal
equivariant quantization. This program, initiated in \cite{Bieliavsky-Gayral},
aims to extend Rieffel's deformation theory \cite{Ri} for more general groups
than
  $\mathbb R^d$. In \cite{Bieliavsky-Gayral}, we have constructed such a theory
for a class of non-Abelian Lie groups. In the present article, we study the
somehow opposite situation of Abelian but non-Lie groups. More specifically, we
construct here a deformation theory of $C^*$-algebras endowed with an action of
a finite dimensional vector space over a non-Archimedean local field of
characteristic different from 2. At the root of our construction stands the
$p$-adic version of the Weyl quantization introduced by Haran and further
extended by
  Bechata and Unterberger.
