  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.
