We present a chemical abundance analysis of four additional confirmed member stars of Tucana III, a Milky Way satellite galaxy candidate in the process of being tidally disrupted as it is accreted by the Galaxy. Two of these stars are centrally located in the core of the galaxy while the other two stars are located in the eastern and western tidal tails. The four stars have chemical abundance patterns consistent with the one previously studied star in Tucana III: They are moderately enhanced in r-process elements, i.e., they have ⟨[Eu/Fe]⟩ ≈ +0.4 dex. The non-neutron-capture elements generally follow trends seen in other dwarf galaxies, including a metallicity range of 0.44 dex and the expected trend in -elements, i.e., the lower metallicity stars have higher Ca and Ti abundances. Overall, the chemical abundance patterns of these stars suggest that Tucana III was an ultra-faint dwarf galaxy, and not a globular cluster, before being tidally disturbed. As is the case for the one other galaxy dominated by r-process enhanced stars, Reticulum II, Tucana III's stellar chemical abundances are consistent with pollution from ejecta produced by a binary neutron star merger, although a different r-process element or dilution gas mass is required to explain the abundances in these two galaxies if a neutron star merger is the sole source of r-process enhancement.