The black hole (BH) binary V404 Cyg entered the outburst phase in 2015 June after 26 yr of X-ray quiescence, and with its behaviour broke the outburst evolution pattern typical of most BH binaries. We observed the entire outburst with the Swift satellite and performed time-resolved spectroscopy of its most active phase, obtaining over a thousand spectra with exposures from tens to hundreds of seconds. All the spectra can be fitted with an absorbed power-law model, which most of the time required the presence of a partial covering. A blueshifted iron-Kα line appears in 10 per cent of the spectra together with the signature of high column densities, and about 20 per cent of the spectra seem to show signatures of reflection. None of the spectra showed the unambiguous presence of soft disc-blackbody emission, while the observed bolometric flux exceeded the Eddington value in 3 per cent of the spectra. Our results can be explained assuming that the inner part of the accretion flow is inflated into a slim disc that both hides the innermost (and brightest) regions of the flow, and produces a cold, clumpy, high-density outflow that introduces the high absorption and fast spectral variability observed. We argue that the BH in V404 Cyg might have been accreting erratically or even continuously at Eddington/super-Eddington rates - thus sustaining a surrounding slim disc - while being partly or completely obscured by the inflated disc and its outflow. Hence, the largest flares produced by the source might not be accretion-driven events, but instead the effects of the unveiling of the extremely bright source hidden within the system.