Neutrino-induced charged- and neutral-current reactions play an important role during and after the r process, if the latter occurs in an environment with extreme neutrino fluxes such as the neutrino-driven wind model or neutron star mergers. Therefore consistent r-process simulations require the knowledge of neutrino-nucleus reaction rates for very neutron-rich nuclei. The neutrino reactions can excite the daughter nucleus above the neutron threshold, which are quite low for r-process nuclei. Thus the daughter nucleus will decay by emission of one or several neutrons. We have calculated the relevant total and partial neutron spallation cross sections for r-process nuclei with neutron numbers $N=41--135.$ Our calculations are based on the random phase approximation and consider allowed as well as forbidden transitions.