Context. Thanks to the relative ease of finding and characterizing small planets around M-dwarf stars, these objects have become cornerstones in the field of exoplanet studies. The current paucity of planets in long-period orbits around M dwarfs makes such objects particularly compelling as they provide clues about the formation and evolution of these systems. Aims. In this study we present the discovery of TOI-2257 b (TIC 198485881), a long-period (35 d) sub-Neptune orbiting an M3 star at 57.8 pc. Its transit depth is about 0.4%, large enough to be detected with medium-size, ground-based telescopes. The long transit duration suggests the planet is in a highly eccentric orbit (e ~ 0.5), which would make it the most eccentric planet known to be transiting an M-dwarf star. Methods. We combined TESS and ground-based data obtained with the 1.0-meter SAINT-EX, 0.60-meter TRAPPIST-North, and 1.2-meter FLWO telescopes to find a planetary size of 2.2 R_⊕ and an orbital period of 35.19 days. In addition, we make use of archival data, high-resolution imaging, and vetting packages to support our planetary interpretation. Results. With its long period and high eccentricity, TOI-2257 b falls into a novel slice of parameter space. Despite the planet’s low equilibrium temperature (~256 K), its host star’s small size (R_* = 0.311 ± 0.015) and relative infrared brightness (K_mag = 10.7) make it a suitable candidate for atmospheric exploration via transmission spectroscopy.