We investigate experimentally the pressure dependence of the production of ultracold neutrons (UCNs) in superfluid helium in the range from saturated vapor pressure to 20 bar. A neutron velocity selector allows the separation of underlying single-phonon and multiphonon processes by varying the incident cold neutron (CN) wavelength in the range from 3.5 to 10 Å. The predicted pressure dependence of UCN production derived from inelastic neutron scattering data is confirmed for the single-phonon excitation. For multiphonon-based UCN production we found no significant dependence on pressure, whereas calculations from inelastic neutron scattering data predict an increase of 43(6)% at 20 bar relative to saturated vapor pressure. From our data we conclude that applying pressure to superfluid helium does not increase the overall UCN production rate at a typical CN guide.