Aims: Our aim is to identify brown dwarf members of the nearby Hyades open star cluster to determine the photometric and spectroscopic properties of brown dwarfs at moderately old ages and extend the knowledge of the substellar mass function of the cluster. Methods: We cross-matched the 2MASS and AllWISE public catalogues and measured proper motions to identify low-mass stars and brown dwarf candidates in an area of radius eight degrees around the central region of the Hyades cluster. We identified objects with photometry and proper motions consistent with cluster membership. For the faintest (J = 17.2 mag) most promising astrometric and photometric low-mass candidate 2MASS J04183483+2131275, with a membership probability of 94.5%, we obtained low-resolution (R = 300-1000) and intermediate-resolution (R = 2500) spectroscopy with the 10.4m Gran Telescopio Canarias. Results: From the low-resolution spectra we determined a L5.0 ± 0.5 spectral type, consistent with the available photometry. In the intermediate dispersion spectrum we detected Hα in emission (marginally resolved with a full width half maximum of 2.8 Å) and determined a log (LHα/Lbol) = -6.0 dex. From Hα we obtained a radial velocity of 38.0 ± 2.9 km s-1, which combined with the proper motion leads to space velocities which are fully consistent with membership in the Hyades cluster. We also report a detection in the H2 band by the UKIDSS Galactic Plane Survey. Using evolutionary models we determine from the available photometry of the object a mass in the range 0.039-0.055 M⊙. Brown dwarfs with masses below 0.055 M⊙ should fully preserve its initial lithium content, and indeed the spectrum at 6708 Å may show a feature consistent with lithium preservation; however, a higher S/N is needed to confirm this point. Conclusions: We have identified a new high-probability L5 brown dwarf member of the Hyades cluster. This is the first relatively old L5 brown dwarf with a well-determined age (500-700 Myr) and measured chromospheric emission. Based on observations made with the Gran Telescopio de Canarias (GTC) installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, on the island of La Palma.