  This article aims to study the Coulomb gas model over the $d$-dimensional
$p$-adic space. We establish the existence of equilibria measures and the
$\Gamma$-limit for the Coulomb energy functional when the number of
configurations tends to infinity. For a cloud of charged particles confined
into the unit ball, we compute the equilibrium measure and the minimum of its
Coulomb energy functional. In the $p$-adic setting the Coulomb energy is the
continuum limit of the minus a hierarchical Hamiltonian attached to a spin
glass model with a $p$-adic coupling.
