  One of the main scientific goals of the TESS mission is the discovery of
transiting small planets around the closest and brightest stars in the sky.
Here, using data from the CARMENES, MAROON-X, and HIRES spectrographs, together
with TESS, we report the discovery and mass determination of a planetary system
around the M1.5 V star GJ 806 (TOI-4481). GJ 806 is a bright (V=10.8 mag, J=7.3
mag) and nearby (d=12 pc) M dwarf that hosts at least two planets. The
innermost planet, GJ 806 b, is transiting and has an ultra-short orbital period
of 0.93 d, a radius of 1.331+-0.023 Re, a mass of 1.90+-0.17 Me, a mean density
of 4.40+-0.45 g/cm3, and an equilibrium temperature of 940+-10 K. We detect a
second, non-transiting, super-Earth planet in the system, GJ 806c, with an
orbital period of 6.6 d, a minimum mass of 5.80+-0.30 Me, and an equilibrium
temperature of 490+-5 K. The radial velocity data also shows evidence for a
third periodicity at 13.6 d, although the current dataset does not provide
sufficient evidence to unambiguously distinguish between a third super-Earth
mass (Msin(i)=8.50+-0.45 Me) planet or stellar activity. Additionally, we
report one transit observation of GJ 806 b taken with CARMENES in search for a
possible extended atmosphere of H or He, but we can only place upper limits to
its existence. This is not surprising as our evolutionary models support the
idea that any possible primordial H/He atmosphere that GJ 806 b might have had,
would long have been lost. However, GJ 806b's bulk density makes it likely that
the planet hosts some type of volatile atmosphere. In fact, with a transmission
spectroscopy metrics (TSM) of 44 and an emission spectroscopy metrics (ESM) of
24, GJ 806 b the third-ranked terrestrial planet around an M dwarf suitable for
transmission spectroscopy studies, and the most promising terrestrial planet
for emission spectroscopy studies.
