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We present the first detailed chemical abundance study of the ultra-faint dwarf galaxy Tucana II, based on high-resolution Magellan/MIKE spectra of four red giant stars. The metallicities of these stars range from [Fe/H] = -3.2 to -2.6, and all stars are low in neutron-capture abundances ([Sr/Fe] and [Ba/Fe] < -1). However, a number of anomalous chemical signatures are present. One star is relatively metal-rich ([Fe/H] = -2.6) and shows [Na, [alpha], Sc/Fe] < 0, suggesting an extended star formation history with contributions from AGB stars and SNe Ia. Two stars with [Fe/H] < -3 are mildly carbon-enhanced ([C/Fe] ~ 0.7) and may be consistent with enrichment by faint supernovae, if such supernovae can produce neutron-capture elements. A fourth star with [Fe/H] = -3 is carbon-normal, and exhibits distinct light element abundance ratios from the carbon-enhanced stars. This carbon-normal star implies that at least two distinct nucleosynthesis sources, both possibly associated with Population III stars, contributed to the early chemical enrichment of this galaxy. Despite its very low luminosity, Tucana II shows a diversity of chemical signatures that preclude it from being a simple "one-shot" first galaxy yet still provide a window into star and galaxy formation in the early universe.