  In this article we study the notion of capacity of a vertex for infinite
graphs over non-Archimedean fields. In contrast to graphs over the real field
monotone limits do not need to exist. Thus, in our situation next to positive
and null capacity there is a third case of divergent capacity. However, we show
that either of these cases is independent of the choice of the vertex and is
therefore a global property for connected graphs. The capacity is shown to
connect the minimization of the energy, solutions of the Dirichlet problem and
existence of a Green's function. We furthermore give sufficient criteria in
form of a Nash-Williams test, study the relation to Hardy inequalities and
discuss the existence of positive superharmonic functions. Finally, we
investigate the analytic features of the transition operator in relation to the
inverse of the Laplace operator.
