ABSTRACT We show that old isolated neutron stars in groups and clusters of galaxies experiencing a Quark-Nova phase (QN: an explosive transition to a quark star) may be the source of fast radio bursts (FRBs). Each of the millions of fragments of the ultrarelativistic QN ejecta provides a collisionless plasma for which the ambient medium (galactic/halo, the intragroup/intracluster medium) acts as a relativistic plasma beam. The Buneman and the Weibel instabilities, successively induced by the beam in the fragment, generate particle bunching and observed coherent emission at GHz frequency with a corresponding fluence in the Jy ms range. The duration, frequency drift, and the rate are in agreement with observed properties of FRBs. Repeats (on time-scales of minutes to months) are due to seeing multiple fragments each beaming at a different direction and coming in at different times. Single (non-repeating) FRBs occur when only emission from the primary fragment is within the detector’s sensitivity. Key properties of FRB 121102 (its years of activity) and of FRB 180916.J0158+65 (its ∼16 d period) are recovered. The spatial and temporal coincidence between SGR 1935+2154 and FRB 200428 finds an explanation in our model. We give testable predictions.