The Milky Way is surrounded by dozens of ultra-faint (< $10^5$ solar luminosities) dwarf satellite galaxies. They are the surviving remnants of the earliest galaxies, as confirmed by their ancient (~13 billion years old) and chemically primitive stars. Simulations suggest that these systems formed within extended dark matter halos and experienced early galaxy mergers and supernova feedback. However, the signatures of these events would lie outside their core regions (>2 half-light radii), which are spectroscopically unstudied due to the sparseness of their distant stars. Here we identify members of the Tucana II ultra-faint dwarf galaxy in its outer region (up to 9 half-light radii), demonstrating the system to be dramatically more spatially extended and chemically primitive than previously found. These distant stars are extremely metal-poor (<[Fe/H]>=-3.02; less than ~1/1000th of the solar iron abundance), affirming Tucana II as the most metal-poor known galaxy. We observationally establish, for the first time, an extended dark matter halo surrounding an ultra-faint dwarf galaxy out to one kiloparsec, with a total mass of >$10^7$ solar masses. This measurement is consistent with the expected ~2x$10^7$ solar masses using a generalized NFW density profile. The extended nature of Tucana II suggests that it may have undergone strong bursty feedback or been the product of an early galactic merger. We demonstrate that spatially extended stellar populations, which other ultra-faint dwarfs hint at hosting as well, are observable in principle and open the possibility for detailed studies of the stellar halos of relic galaxies.
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