We report the discovery and characterisation of a giant transiting planet orbiting a nearby M3.5V dwarf (d = 80.4pc, G = 15.1 mag, K=11.2mag, R* = 0.358 ± 0.015 R⊙, M* = 0.340 ± 0.009 M⊙). Using the photometric time series from TESS sectors 10, 36, 46, and 63 and near-infrared spectrophotometry from ExTrA, we measured a planetary radius of 0.77 ± 0.03 RJ and an orbital period of 1.52 days. With high-resolution spectroscopy taken by the CFHT/SPIRou and ESO/ESPRESSO spectrographs, we refined the host star parameters ([Fe/H] = 0.27 ± 0.12) and measured the mass of the planet (0.273 ± 0.006 MJ). Based on these measurements, TOI-4860 b joins the small set of massive planets (>80 ME) found around mid to late M dwarfs (<0.4 R⊙), providing both an interesting challenge to planet formation theory and a favourable target for further atmospheric studies with transmission spectroscopy. We identified an additional signal in the radial velocity data that we attribute to an eccentric planet candidate (e = 0.66 ± 0.09) with an orbital period of 427 ± 7 days and a minimum mass of 1.66 ± 0.26 MJ, but additional data would be needed to confirm this.