The self-diffusion coefficients (D) of the cation and anion in the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM]PF6) have been determined together with the electrical conductivity (kappa) under high pressure. All three quantities strongly decrease with increasing pressure to similar to 20% of their atmospheric pressure values at 200 MPa. D(PF6-) is always less than D([BMIM](+)), despite the larger van der Waals volume of the cation. The pressure effect on the transport coefficients is discussed in terms of velocity correlation coefficients (VCCs or f(ij)), the Nernst-Einstein equation (ionic diffusivity-conductivity), and the fractional form of the Stokes-Einstein relation (viscosity-conductivity and viscosity-diffusivity). It was found that the VCCs for the cation-cation, anion-anion, and cation-anion pairs are all negative and strongly pressure-dependent, increasing (becoming less negative) with increasing pressure. However, when the values of the VCCs for a given isotherm are normalized relative to the corresponding atmospheric pressure values, they collapse onto a single curve, as might be expected because the pressure should affect the interionic velocity correlations in the same way for each type of interaction. These isothermal curves can be represented by the form exp(alpha p + beta p(2)). The Nernst-Einstein deviation parameter, Delta, which depends on the differences between the like-like ion and unlike ion VCCs (f(++) + f(--) - 2f(+-)), is very nearly constant under the conditions examined. The diffusion and molar conductivity (Lambda) data are found to fit fractional forms of the Stokes-Einstein relationship with the viscosity, (Lambda T) proportional to (T/eta)(t) and D-i proportional to (T/eta)(t) , with t = (0.92 +/- 0.05), independent of both temperature and pressure within the ranges studied and common to the three independently determined properties.