The GAPS programme at TNG XLIII. A massive brown dwarf orbiting the active M dwarf TOI-5375

Context. Massive substellar companions orbiting active low-mass stars are rare. They, however, offer an excellent opportunity to study the main mechanisms involved in the formation and evolution of substellar objects. Aims. We aim to unravel the physical nature of the transit signal observed by the TESS space mission on the active M dwarf TOI-5375. Methods. We analysed the available TESS photometric data as well as high-resolution (R $\sim$ 115000) HARPS-N spectra. We combined these data to characterise the star TOI-5375 and to disentangle signals related to stellar activity from the companion transit signal in the light-curve data. We ran an MCMC analysis to derive the orbital solution and apply state-of-the-art Gaussian process regression to deal with the stellar activity signal. Results. We reveal the presence of a companion in the brown dwarf / very-low-mass star boundary orbiting around the star TOI-5375. The best-fit model corresponds to a companion with an orbital period of 1.721564 $\pm$ 10$^{\rm -6}$ d, a mass of 77 $\pm$ 8 $M_{\rm J}$ and a radius of 0.99 $\pm$ 0.16 $R_{\rm J}$. Conclusions. We derive a rotation period for the host star of 1.9692 $\pm$ 0.0004 d, and we conclude that the star is very close to synchronising its rotation with the orbital period of the companion.