GRB160709A is one of the small sample of short Gamma-ray Bursts detected by both the Gamma-ray Burst Monitor and the Large Area Telescope on-board the \textit{Fermi} Gamma-ray Space Telescope. We performed a detailed spectral analysis of the Fermi gamma-ray data for this GRB. The spectrum is well described by a combination of thermal and non-thermal spectral components. The lightcurve of this GRB can be described by several phases: a weak precursor, an intense main peak and a weak tail. In a time-resolved analysis of the main peak, we found evidence of a period dominated by thermal emission with a temperature of $\sim$340 keV. In addition to the spectral components dominating the low energy gamma-ray range, a hard power-law component with constant photon index is required to adequately describe the data at high energies. This extra component is present from the main peak through the weak tail. The flux of this component decreases with time, similar to the behavior seen in the bright short GRB090510