Context. Although more than 2000 brown dwarfs have been detected to date, mainly from direct imaging, their characterisation is difficult due to their faintness and model-dependent results. In the case of transiting brown dwarfs, however, it is possible to make direct high-precision observations. Aims. Our aim is to investigate the nature and formation of brown dwarfs by adding a new well-characterised object, in terms of its mass, radius and bulk density, to the currently small sample of less than 20 transiting brown dwarfs. Methods. One brown dwarf candidate was found by the KESPRINT consortium when searching for exoplanets in the K2 space mission Campaign 16 field. We combined the K2 photometric data with a series of multicolour photometric observations, imaging, and radial velocity measurements to rule out false positive scenarios and to determine the fundamental properties of the system. Results. We report the discovery and characterisation of a transiting brown dwarf in a 5.17-day eccentric orbit around the slightly evolved F7 V star EPIC 212036875. We find a stellar mass of 1.15 ± 0.08 M_⊙, a stellar radius of 1.41 ± 0.05 R_⊙, and an age of 5.1 ± 0.9 Gyr. The mass and radius of the companion brown dwarf are 51 ± 2 M_J and 0.83 ± 0.03 R_J, respectively, corresponding to a mean density of 108₋₁₃⁺¹⁵ g cm⁻³. Conclusions. EPIC 212036875 b is a rare object that resides in the brown-dwarf desert. In the mass-density diagram for planets, brown dwarfs, and stars, we find that all giant planets and brown dwarfs follow the same trend from ~0.3 M_J to the turn-over to hydrogen burning stars at ~ 73 M_J. EPIC 212036875 b falls close to the theoretical model for mature H/He dominated objects in this diagram as determined by interior structure models. We argue that EPIC 212036875 b formed via gravitational disc instabilities in the outer part of the disc, followed by a quick migration. Orbital tidal circularisation may have started early in its history for a brief period when the brown dwarf’s radius was larger. The lack of spin–orbit synchronisation points to a weak stellar dissipation parameter (Q_⋆^′ ≳ 10⁸), which implies a circularisation timescale of ≳23 Gyr, or suggests an interaction between the magnetic and tidal forces of the star and the brown dwarf.