Long gamma-ray bursts mark the death of massive stars, as revealed by their association with energetic broad-lined stripped-envelope supernovae. The scarcity of nearby events and the brightness of the GRB afterglow, dominating the first days of emission, have so far prevented the study of the very early stages of the GRB-SN evolution. Here we present detailed, multi-epoch spectroscopic observations of SN 2017iuk, associated with GRB 171205A which display features at extremely high expansion velocities of $sim$ 100,000 km s$^{-1}$ within the first day after the burst. These high-velocity components are characterized by chemical abundances different from those observed in the ejecta of SN 2017iuk at later times. Using spectral synthesis models developed for the SN 2017iuk, we explain these early features as originating not from the supernova ejecta, but from a hot cocoon generated by the energy injection of a mildly-relativistic GRB jet expanding into the medium surrounding the progenitor star. This cocoon becomes rapidly transparent and is outshone by the supernova emission which starts dominating three days after the burst. These results proves that the jet plays an important role not only in powering the GRB event but also its associated supernova.