Dynamical histories of planetary systems, as well as the atmospheric evolution of highly irradiated planets, can be studied by characterizing the ultra-short-period planet population, which the TESS mission is particularly well suited to discover. Here, we report on the follow-up of a transit signal detected in the TESS sector 19 photometric time series of the M3.0 V star TOI-1685 (2MASS J04342248+4302148). We confirm the planetary nature of the transit signal, which has a period of Pb = 0.6691403−0.0000021+0.0000023 d, using precise radial velocity measurements taken with the CARMENES spectrograph. From the joint photometry and radial velocity analysis, we estimate the following parameters for TOI-1685 b: a mass of Mb = 3.78−0.63+0.63 M⊕, a radius of Rb = 1.70−0.07+0.07 R⊕, which together result in a bulk density of ρb = 4.21−0.82+0.95 g cm−3, and an equilibrium temperature of Teq = 1069−16+16 K. TOI-1685 b is the least dense ultra-short-period planet around an M dwarf known to date. TOI-1685 b is also one of the hottest transiting super-Earth planets with accurate dynamical mass measurements, which makes it a particularly attractive target for thermal emission spectroscopy. Additionally, we report with moderate evidence an additional non-transiting planet candidate in the system, TOI-1685 [c], which has an orbital period of Pc = 9.02−0.12+0.10 d.