Radiogenic neutrons emitted by detector materials are one of the most challenging backgrounds for the direct search of dark matter in the form of weakly interacting massive particles (WIMPs). To mitigate this background, the XENONnT experiment is equipped with a novel gadolinium-doped water Cherenkov detector, which encloses the xenon dual-phase time projection chamber (TPC). The neutron veto (NV) can tag neutrons via their capture on gadolinium or hydrogen, which release γ-rays that are subsequently detected as Cherenkov light. In this work, we present the first results of the XENONnT NV when operated with demineralized water only, before the insertion of gadolinium. Its efficiency for detecting neutrons is (82 ± 1)%, the highest neutron detection efficiency achieved in a water Cherenkov detector. This enables a high efficiency of (53 ± 3)% for the tagging of WIMP-like neutron signals, inside a tagging time window of 250 μs between TPC and NV, leading to a livetime loss of 1.6% during the first science run of XENONnT.