We report the discovery of HATS-71b, a transiting gas giant planet on a P = 3.7955 day orbit around a G = 15.35 mag M3 dwarf star. HATS-71 is the coolest M dwarf star known to host a hot Jupiter. The loss of light during transits is 4.7%, more than any other confirmed transiting planet system. The planet was identified as a candidate by the ground-based HATSouth transit survey. It was confirmed using ground-based photometry, spectroscopy, and imaging, as well as space-based photometry from the NASA TESS mission (TIC 234523599). Combining all of these data, and utilizing Gaia DR2, we find that the planet has a radius of $1.080 pm 0.016 R_J$ and mass of $0.45 pm 0.24 M_J$ (95% confidence upper limit of $0.81 M_J$ ), while the star has a mass of $0.569 pm^{0.042}_{0.069},M_odot$ and a radius of $0.5161pm^{0.0053}_{0.0099},R_odot$. The Gaia DR2 data show that HATS-71 lies near the binary main sequence in the Hertzsprung-Russell diagram, suggesting that there may be an unresolved stellar binary companion. All of the available data is well fitted by a model in which there is a secondary star of mass $0.24 M_odot$, although we caution that at present there is no direct spectroscopic or imaging evidence for such a companion. Even if there does exist such a stellar companion, the radius and mass of the planet would be only marginally different from the values we have calculated under the assumption that the star is single.
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