  We report the confirmation and mass determination of a mini-Neptune
transiting the M3.5 V star TOI-4438 (G 182-34) every 7.44 days. A transit
signal was detected with NASA's TESS space mission in the sectors 40, 52, and
53. In order to validate the planet TOI-4438 b and to determine the system
properties, we combined TESS data with high-precision radial velocity
measurements from the CARMENES spectrograph, spanning almost one year, and
ground-based transit photometry. We found that TOI-4438 b has a radius of Rb =
2.52 +/- 0.13 R_Earth (5% precision), which together with a mass of Mb=5.4 +/-
1.1 M_Earth (20% precision), results in a bulk density of rho = 1.85+0.51-0.44
g cm-3 (28% precision), aligning the discovery with a volatile-rich planet. Our
interior structure retrieval with a pure water envelope yields a minimum water
mass fraction of 46% (1-sigma). TOI-4438 b is a volatile-rich mini-Neptune with
likely H/He mixed with molecules, such as water, CO_2, and CH_4. The primary
star has a J-band magnitude of 9.7, and the planet has a high transmission
spectroscopy metric (TSM) of 136 +/- 13. Taking into account the relatively
warm equilibrium temperature of T_eq = 435 +/- 15 K, and the low activity level
of its host star, TOI-4438 b is one of the most promising mini-Neptunes around
an M dwarf for transmission spectroscopy studies.
