Nearly all glass-forming liquids feature, along with the structural alpha-relaxation process, a faster secondary process (beta relaxation), whose nature belongs to the great mysteries of glass physics. However, for some of these liquids, no well-pronounced secondary relaxation is observed. A prominent example is the archetypical glass-forming liquid glycerol. In the present work, by performing dielectric spectroscopy under superhigh pressures up to 6 GPa, we show that in glycerol a significant secondary relaxation peak appears in the dielectric loss at P>3 GPa. We identify this beta relaxation to be of Johari-Goldstein type and discuss its relation to the excess wing. We provide evidence for a smooth but significant increase in glass-transition temperature and fragility on increasing pressure.