In this paper we present a new formalism to implement the nuclear particle-vibration coupling (PVC) model. The key issue is the proper treatment of the continuum that is allowed by the coordinate space representation. Our formalism, based on the use of zero-range interactions such as the Skyrme forces, is microscopic and fully self-consistent. We apply it to the case of neutron single-particle states in 40 Ca, 208 Pb, and 24 O. The first two cases are meant to illustrate the comparison with the usual (i.e., discrete) PVC model. However, we stress that the present approach allows one to calculate properly the effect of PVC on resonant states. We compare our results with those from experiments in which the particle transfer in the continuum region has been attempted. The latter case, namely 24 O, is chosen as an example of a weakly-bound system. Such a nucleus, being double magic and not displaying collective low-lying vibrational excitations, is characterized by quite pure neutron single-particle states around the Fermi surface.