GRB 160709A is one of the few bright short gamma-ray bursts detected by both the Gamma-ray Burst Monitor and the Large Area Telescope on board the Fermi Gamma-ray Space Telescope. The γ-ray prompt emission of GRB 160709A is adequately fitted by combinations of three distinct components: (i) a nonthermal component described by a power law (PL) with a high-energy exponential cutoff, (ii) a thermal component modeled with a Planck function, and (iii) a second nonthermal component shaped by an additional PL crossing the whole γ-ray spectrum. While the thermal component dominates during ∼0.12 s of the main emission episode of GRB 160709A with an unusually high temperature of ∼340 keV, the nonthermal components dominate in the early and late time. The thermal component is consistent with the photospheric emission resulting in the following parameters: the size of the central engine, × 108 cm, the size of the photosphere, × 1010 cm, and a bulk Lorentz factor, , assuming a redshift of 1. The slope of the additional PL spectrum stays unchanged throughout the burst duration; however, its flux decreases continuously as a function of time. A standard external shock model has been tested for the additional PL component using the relation between the temporal and spectral indices (the closure relation). Each set of spectral and temporal indices from two energy bands (200 keV-40 MeV and 100 MeV-10 GeV) satisfies a distinct closure relation. From the closure relation test we derived the index for the electron spectral distribution, p = 2.5 0.1. The interaction of the jet with the interstellar environment is preferred over the interaction with the wind medium.