Context. Type Ic supernovae (SNe Ic) arise from the core-collapse of H-(and He-) poor stars, which could either be single Wolf-Rayet (WR) stars or lower-mass stars stripped of their envelope by a companion. Their light curves are radioactively powered and usually show a fast rise to peak (∼10−15 d), without any early (in the first few days) emission bumps (with the exception of broad-lined SNe Ic) as sometimes seen for other types of stripped-envelope SNe (e.g., Type IIb SN 1993J and Type Ib SN 2008D). Aims. We have studied iPTF15dtg, a spectroscopically normal SN Ic with an early excess in the optical light curves followed by a long (∼30 d) rise to the main peak. It is the first spectroscopically-normal double-peaked SN Ic to be observed. Our aim is to determine the properties of this explosion and of its progenitor star. Methods. Optical photometry and spectroscopy of iPTF15dtg was obtained with multiple telescopes. The resulting light curves and spectral sequence are analyzed and modeled with hydrodynamical and analytical models, with particular focus on the early emission. Results. iPTF15dtg is a slow rising SN Ic, similar to SN 2011bm. Hydrodynamical modeling of the bolometric properties reveals a large ejecta mass (∼10 M ) and strong 56 Ni mixing. The luminous early emission can be reproduced if we account for the presence of an extended ( 500 R ), low-mass ( 0.045 M ) envelope around the progenitor star. Alternative scenarios for the early peak, such as the interaction with a companion, a shock-breakout (SBO) cooling tail from the progenitor surface, or a magnetar-driven SBO are not favored. Conclusions. The large ejecta mass and the presence of H-and He-free extended material around the star suggest that the progenitor of iPTF15dtg was a massive ( 35 M ) WR star that experienced strong mass loss.