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Aims: Our objective is the optical and near-infrared spectroscopic characterisation of 2MASS J0249-0557 c, a recently discovered young planetary mass companion to the $β$ Pictoris member 2MASS J0249-0557. Methods: Using the Visible and Infrared Survey Telescope for Astronomy (VISTA) Hemisphere Survey (VHS) and the Two Micron All Sky Survey (2MASS) data, we independently identified the companion 2MASS J0249-0557 c. We obtained low-resolution optical spectroscopy of this object using the Optical System for Imaging and low-intermediate-Resolution Integrated Spectroscopy (OSIRIS) spectrograph at the Gran Telescopio Canarias (GTC), and near-infrared spectroscopy using the Son of Isaac (SofI) spectrograph on the New Technology Telescope (NTT). Results: We classified 2MASS J0249-0557 c with a spectral type of L2.5$\pm$0.5 in the optical and L3$\pm$1 in the near-infrared. We identified spectroscopic indicators of youth that are compatible with the age of the $β$ Pictoris moving group. We also detect a strong H$α$ emission, with a pEW of -90$^{+20}_{-40}$A, which seems persistent in time. This indicates strong chromospheric activity or disk accretion. Although many M-type brown dwarfs have strong H$α$ emission, this target is one of the very few L-type planetary mass objects in which this strong H$α$ emission has been detected. Lithium absorption at 6708 A is observed with pEW $\lesssim$ 5A. We also computed the binding energy of 2MASS J0249-0557 c and obtained an (absolute) upper limit of $U=(-8.8\pm4.4) 10^{32}$ J. Conclusions: Similarly to other young brown dwarfs and isolated planetary mass objects, strong H$α$ emission is also present in young planetary mass companions at ages of some dozen million years. We also found that 2MASS J0249-0557 c is one of the wide substellar companions with the lowest binding energy known to date.