The properties of the low-energy dipole response are investigated for the proton-rich doubly magic nucleus 48Ni, in a comparative study of two microscopic models: fully self-consistent relativistic random-phase approximation (RRPA) based on the novel density-dependent meson-exchange interactions, and continuum random-phase approximation (CRPA) using Skyrme-type interactions with the continuum properly included. Both models predict the existence of the low-energy soft mode, i.e., the proton pygmy dipole resonance (PDR), for which the transition densities and RPA amplitudes indicate the dynamics of loosely bound protons vibrating against the rest of the nucleons. The CRPA analysis indicates that the escape width for the proton PDR is rather large, as a result of the coupling to the continuum.