The conductivity relaxation dynamics of the room-temperature ionic liquid 1-methyl-3-trimethylsilylmethylimidazolium tetrafluoroborate ([Si-MIm][BF(4)]) have been studied by broadband conductivity relaxation measurements at ambient pressure and elevated pressures up to 600 MPa. For the first time, several novel features of the dynamics have been found in a room-temperature ionic liquid. In the electric loss modulus M″(f) spectra, a resolved secondary β-conductivity relaxation appears, and its relaxation time τ(β) shifts on applying pressure in concert with the relaxation time τ(α) of the primary α-conductivity relaxation. The spectral dispersion of the α-conductivity relaxation, as well as the fractional exponent (1 - n) of the Kohlrausch-Williams-Watts function that fits the spectral dispersion, is invariant to various combinations of pressure and temperature that keep τ(α) constant. Moreover, τ(β) is unchanged. Thus the three quantities, τ(α), τ(β), and n, are coinvariant to changes in pressure and temperature. This strong connection to the α-conductivity relaxation shown by the β-conductivity relaxation in [Si-MIm][BF(4)] indicates that it is the analogue of the Johari-Goldstein β-relaxation in nonionically conducting glass-formers. The findings have fundamental implications on theoretical interpretation of the conductivity relaxation processes and glass transition in ionic liquids. It is also the first time such a secondary conductivity relaxation or the primitive conductivity relaxation of the coupling model has been fully resolved and identified in M″(f) in any ionically conducting material that we know of.