Context. KQ Vel is a peculiar A0p star with a strong surface magnetic field of about 7.5 kG. It has a slow rotational period of nearly 8 years. Bailey et al. (AandA, 575, A115) detected a binary companion of uncertain nature and suggested that it might be a neutron star or a black hole. Aims. We analyze X-ray data obtained by the Chandra telescope to ascertain information about the stellar magnetic field and/or interaction between the star and its companion. Methods. We confirm previous X-ray detections of KQ Vel with a relatively high X-ray luminosity of 2 × 1030 erg s-1. The X-ray spectra suggest the presence of hot gas at > 20 MK and, possibly, of a nonthermal component. The X-ray light curves are variable, but data with better quality are needed to determine a periodicity, if any. Results. We interpret the X-ray spectra as a combination of two components: the nonthermal emission arising from the aurora on the A0p star, and the hot thermal plasma filling the extended shell that surrounds the "propelling"neutron star. Conclusions. We explore various alternatives, but a hybrid model involving the stellar magnetosphere along with a hot shell around the propelling neutron star seems most plausible. We speculate that KQ Vel was originally a triple system and that the Ap star is a merger product. We conclude that KQ Vel is an intermediate-mass binary consisting of a strongly magnetic main-sequence star and a neutron star.