Pump-probe photoelectron spectroscopy (PES) is a versatile tool to investigate the dynamics of transient states of excited matter. Vacuum space-charge effects can mask these dynamics and complicate the interpretation of electron spectra. Here we report on space-charge effects in Au 4f photoemission from a polycrystalline gold surface, excited with moderately intense 90 ps (FWHM) soft x-ray probe pulses, under the influence of the Coulomb forces exerted by a pump electron cloud, which was produced by intense 40 fs laser pulses. The experimentally observed kinetic energy shift and spectral broadening of the Au 4f lines, measured with highly-efficient time-of-flight spectroscopy, are in good agreement with simulations utilizing a mean-field model of the electrostatic pump electron potential. This confirms that the line broadening is predominantly caused by variations in the take-off time of the probe electrons without appreciable influence of local scattering events. Our findings might be of general interest for pump-probe PES with picosecond-pulse-length sources.