A pseudopotential approach is used to study the lattice and elastic properties of the wide band-gap GaN at zero and high pressures up to 120 kbar. When the pressure is 0 kbar, our findings are generally in agreement with the data reported in the literature. The pressure dependence of lattice constant, polarity, transverse effective charge, elastic constants and their related mechanical parameters, and microhardness has been examined and discussed. Our results show that all these features exhibit a monotonic behaviour against pressure. Upon compression up to 120 kbar, our results suggest that the material in question remains mechanically stable with higher stiffness, becomes more resistant to the deformations or deflections and its chemical bond and rigidity become stronger.