The article reports on a molecular dynamics simulation study of the influence of moderate, nondenaturing pressure on the slow and fast internal relaxation dynamics of lysozyme. The model parameters of the fractional Ornstein-Uhlenbeck process are used to quantify the changes. We find that the nonexponential character for diffusive motions on time scales above 10 ps is enhanced and that the diffusion processes are slowed down. The diffusive motions on the subpicosecond time scale appear, in contrast, accelerated, whereas the nonexponential character is not altered by pressure. We attribute these findings to the different natures of slow and fast relaxation processes, which are characterized by structural rearrangements and collisions, respectively. The analyses are facilitated by the use of spatially resolved relaxation rate spectra.