We present the detection of persistent soft X-ray radiation with L_x_~10^41^-10^42^erg/s at the location of the extremely luminous, double-humped transient ASASSN-15lh as revealed by Chandra and Swift. We interpret this finding in the context of observations from our multiwavelength campaign, which revealed the presence of weak narrow nebular emission features from the host-galaxy nucleus and clear differences with respect to superluminous supernova optical spectra. Significant UV flux variability on short timescales detected at the time of the rebrightening disfavors the shock interaction scenario as the source of energy powering the long-lived UV emission, while deep radio limits exclude the presence of relativistic jets propagating into a low-density environment. We propose a model where the extreme luminosity and double-peaked temporal structure of ASASSN-15lh is powered by a central source of ionizing radiation that produces a sudden change in the ejecta opacity at later times. As a result, UV radiation can more easily escape, producing the second bump in the light curve. We discuss different interpretations for the intrinsic nature of the ionizing source. We conclude that, if the X-ray source is physically associated with the optical-UV transient, then ASASSN-15lh most likely represents the tidal disruption of a main-sequence star by the most massive spinning black hole detected to date. In this case, ASASSN-15lh and similar events discovered in the future would constitute the most direct probes of very massive, dormant, spinning, supermassive black holes in galaxies. Future monitoring of the X-rays may allow us to distinguish between the supernova hypothesis and the hypothesis of a tidal disruption event.