Point-contact p-type high-purity germanium detectors (PPC HPGe) are particularly suited for detection of sub-keV nuclear recoils from coherent elastic scattering of neutrinos or light dark matter particles. While these particles are expected to interact homogeneously in the entire detector volume, specific classes of external background radiation preferably deposit their energy close to the semi-active detector surface, in which diffusion processes dominate that subsequently lead to slower rising pulses compared to the ones from the fully active bulk volume. Dedicated studies of their shape are therefore highly beneficial for the understanding and the rejection of these unwanted events. This article reports about the development of a data-driven pulse shape discrimination (PSD) method for the four 1 kg size PPC HPGe detectors of the CONUS experiment in the keV and sub-keV regime down to 210 eVee. The impact of the electronic noise at such low energies is carefully examined. It is shown that for an acceptance of 90% of the faster signal-like pulses from the bulk volume, approx. 50% of the surface events can be rejected at the energy threshold and that their contribution is fully suppressed above 800 eVee. Applied to the CONUS background data, such a PSD rejection cut allows to achieve an overall (15-25)% reduction of the total background budget. The new method allows to improve the sensitivity of future CONUS analyses and to refine the corresponding background model in the sub-keV energy region.