We report a study of the effects of pressure on the diffusivity of water molecules confined in single- wall carbon nanotubes (SWNT) with average mean pore diameter of 16 Angstroms. The measurements were carried out using high-resolution neutron scattering, over the temperature range 220 < T < 260 K, and at two pressure conditions: ambient and elevated pressure. The high pressure data were collected at constant volume on cooling, with P varying from 1.92 kbar at temperature T = 260 K to 1.85 kbar at T = 220 K. Analysis of the observed dynamic structure factor S(Q, E) reveals the presence of two relaxation processes, a faster diffusion component (FC) associated with the motion of caged or restricted molecules, and a slower component arising from the free water molecules diffusing within the SWNT matrix. While the temperature dependence of the slow relaxation time exhibits a Vogel-Fulcher-Tammann law and is non-Arrhenius in nature, the faster component follows an Arrhenius exponential law at both pressure conditions. The application of pressure remarkably slows down the overall molecular dynamics, in agreement with previous observations, but most notably affects the slow relaxation. The faster relaxation shows marginal or no change with pressure within the experimental conditions.