The recent aLIGO/aVirgo discovery of gravitational waves from the neutron star merger (NSM) GW170817 and the follow-up kilonova observations have shown that NSMs produce copious amounts of r -process material. However, it is difficult to reconcile the large natal kicks and long average merging times of double-neutron stars (DNSs) with the levels of r -process enrichment seen in ultra-faint dwarf (UFD) galaxies such as Reticulum II and Tucana III. Assuming that such dwarf systems have lost a significant fraction of their stellar mass through tidal stripping, we conclude that contrary to most current models, it is the DNSs with rather large natal kicks but very short merging timescales that can enrich UFD-type galaxies. These binaries are either on highly eccentric orbits or form with very short separations due to an additional mass transfer between the first-born neutron star and a naked helium star, the progenitor of the second neutron star. These DNSs are born with a frequency that agrees with the statistics of the r -process UFDs, and merge well within the virial radius of their host halos, therefore contributing significantly to their r -process enrichment.