The pressure induced quantum phase transition of the weakly ferromagnetic metal MnSi is studied using zero-field 29Si NMR spectroscopy and relaxation. Below P(*) approximately 1.2 GPa, the intensity of the signal and the nuclear spin-lattice relaxation are independent of pressure, even though the amplitude of the magnetization drops by 20% from the ambient-pressure amplitude. For P>P(*), the decreasing intensity within the experimentally detectable bandwidth signals the onset of an inhomogeneous phase that persists to the highest pressure measured, P>/=1.75 GPa, which is well beyond the known critical pressure P(c)=1.46 GPa. Implications for the non-Fermi liquid behavior observed for P>P(c) are discussed.