We present a pressure-dependence study of the dynamics of lysozyme protein powder immersed in deuterated α,α-trehalose environment via quasielastic neutron scattering (QENS). The goal is to assess the baroprotective benefits of trehalose on biomolecules by comparing the findings with those of a trehalose-free reference study. While the mean-square displacement of the trehalose-free protein (hydrated to dD2O≃40 w%) as a whole, is reduced by increasing pressure, the actual observable relaxation dynamics in the picoseconds to nanoseconds time range remains largely unaffected by pressure--up to the maximum investigated pressure of 2.78(2) Kbar. Our observation is independent of whether or not the protein is mixed with the deuterated sugar. This suggests that the hydrated protein's conformational states at atmospheric pressure remain unaltered by hydrostatic pressures, below 2.78 Kbar. We also found the QENS response to be totally recoverable after ambient pressure conditions are restored. Small-angle neutron diffraction measurements confirm that the protein-protein correlation remains undisturbed. We observe, however, a clear narrowing of the QENS response as the temperature is decreased from 290 to 230 K in both cases, which we parametrize using the Kohlrausch-Williams-Watts stretched exponential model. Only the fraction of protons that are immobile on the accessible time window of the instrument, referred to as the elastic incoherent structure factor, is observably sensitive to pressure, increasing only marginally but systematically with increasing pressure.