  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
$0.5161\pm^{0.0053}_{0.0099}\,R_\odot$. 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.
