We report experimental evidence for pressure instabilities in the model multiferroic BiFeO3 and namely reveal two structural phase transitions around 3 GPa and 10 GPa by using diffraction and far-infrared spectroscopy at a synchrotron source. The intermediate phase from 3 to 9 GPa crystallizes in a monoclinic space group, with octahedra tilts and small cation displacements. When the pressure is further increased the cation displacements (and thus the polar character) of BiFeO3 is suppressed above 10 GPa. The above 10 GPa observed non-polar orthorhombic Pnma structure is in agreement with recent theoretical ab-initio prediction, while the intermediate monoclinic phase was not predicted theoretically.