The authors define Feynman integrals with \(p\)-adic arguments of a complex dimension. Such an integral \(F_G(y,\lambda ,d)\) (determined by a graph \(G\)) depends on a \(p\)-adic vector \(y\), a collection \(\lambda\) of complex parameters, and the dimension \(d\). It is shown that \(F_G\) is meromorphic with respect to \(\lambda\) and \(d\) for a fixed \(y\). Then it is considered as a distribution in \(y\) which admits an analytic continuation with respect to \(d\); poles of the extended function are studied. Their location depends on the structure of \(G\). This makes it possible to carry out dimensional renormalization for the \(p\)-adic case. For the fermionic model another procedure of dimensional renormalization is given, based on renormalization group properties of a hierarchical model which provides a discretization for the \(p\)-adic quantum field model.