The ejection of neutron-rich matter is one of the most important consequences of a neutron star merger. While the bulk of the matter is ejected at fast, but non-relativistic velocities (similar to 0.2c), a small amount of mildly relativistic dynamic ejecta have been seen in a number of numerical simulations. Such ejecta can have far-reaching observational consequences ranging from the shock breakout burst of gamma-rays promptly after the merger, to an early (similar to 1 h post-merger) blue kilonova precursor signal, to synchrotron emission years after the merger ('kilonova afterglow'). These all potentially carry the imprint of the binary system parameters and the equation of state. By analysing Lagrangian simulations in full general relativity, performed with the code sphincs_bssn, we identify two ejection mechanisms for fast ejecta: (i) about 30 per cent of the ejecta with v>0.4c are 'sprayed out' from the shear interface between the merging stars and escape along the orbital plane and (ii) the remaining similar to 70 per cent of the fast ejecta result from the central object 'bouncing back' after strong, general-relativistic compression. This 'bounce component' is ejected in a rather isotropic way and reaches larger velocities (by similar to 0.1c) so that its faster parts can catch up with and shock slower parts of the spray ejecta. Even for a case that promptly collapses to a black hole, we find fast ejecta with similar properties to the non-collapsing case, while slower matter parts are swallowed by the forming black hole. We discuss observational implications of these fast ejecta, including shock breakout and kilonova afterglow.