Cosmological gamma-ray bursts (GRBs) are known to arise from distinct progenitor channels: short GRBs mostly from neutron star mergers and long GRBs from a rare type of core-collapse supernova (CCSN) called collapsars. Highly magnetized neutron stars called magnetars also generate energetic, short-duration gamma-ray transients called magnetar giant flares (MGFs). Three have been observed from the Milky Way and its satellite galaxies, and they have long been suspected to constitute a third class of extragalactic GRBs. We report the unambiguous identification of a distinct population of four local (<5 Mpc) short GRBs, adding GRB 070222 to previously discussed events. While identified solely based on alignment with nearby star-forming galaxies, their rise time and isotropic energy release are independently inconsistent with the larger short GRB population at >99.9% confidence. These properties, the host galaxies, and non-detection in gravitational waves all point to an extragalactic MGF origin. Despite the small sample, the inferred volumetric rates for events above 4 × 10(exp 44) erg of R(sub MGF) =3.8(sup +4.0)(sub -3.1) ×10(exp 5) Gpc(exp -3) yr(exp -1) make MGFs the dominant gamma-ray transient detected from extragalactic sources. As previously suggested, these rates imply that some magnetars produce multiple MGFs, providing a source of repeating GRBs. The rates and host galaxies favor common CCSN as key progenitors of magnetars.