The connection between short gamma-ray bursts (SGRBs) and binary neutron star (BNS) mergers was recently confirmed by the association of GRB 170817A with the merger event GW170817. However, no conclusive indications were obtained on whether the merger remnant that powered the SGRB jet was an accreting black hole (BH) or a long-lived massive neutron star (NS). Here, we explore the latter case via BNS merger simulations covering up to 250 ms after merger. We report, for the first time in a full merger simulation, the formation of a magnetically driven collimated outflow along the spin axis of the NS remnant. For the system at hand, the properties of such an outflow are found largely incompatible with an SGRB jet. With due consideration of the limitations and caveats of our present investigation, our results favour a BH origin for GRB 170817A and SGRBs in general. Even though this conclusion needs to be confirmed by exploring a larger variety of physical conditions, we briefly discuss possible consequences of all SGRB jets being powered by accreting BHs.