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We report the discovery and characterization of TOI-1759~b, a temperate (400 K) sub-Neptune-sized exoplanet orbiting the M~dwarf TOI-1759 (TIC 408636441). TOI-1759 b was observed by TESS to transit on sectors 16, 17 and 24, with only one transit observed per sector, creating an ambiguity on the orbital period of the planet candidate. Ground-based photometric observations, combined with radial-velocity measurements obtained with the CARMENES spectrograph, confirm an actual period of $18.85019 \pm 0.00014$ d. A joint analysis of all available photometry and radial velocities reveal a radius of $3.17 \pm 0.10\,R_\oplus$ and a mass of $10.8 \pm 1.5\,M_\oplus$. Combining this with the stellar properties derived for TOI-1759 ($R_\star = 0.597 \pm 0.015\,R_\odot$; $M_\star = 0.606 \pm 0.020\,M_\odot$; $T_{\textrm{eff}} = 4065 \pm 51$ K), we compute a transmission spectroscopic metric (TSM) value of over 80 for the planet, making it a good target for transmission spectroscopy studies. TOI-1759 b is among the top five temperate, small exoplanets ($T_\textrm{eq} < 500$ K, $R_p < 4 \,R_\oplus$) with the highest TSM discovered to date. Two additional signals with periods of 80 d and $>$ 200 d seem to be present in our radial velocities. While our data suggest both could arise from stellar activity, the later signal's source and periodicity are hard to pinpoint given the $\sim 200$ d baseline of our radial-velocity campaign with CARMENES. Longer baseline radial-velocity campaigns should be performed in order to unveil the true nature of this long period signal.