Understanding the high-pressure strength of materials is vital to developing physics-based material models for high-velocity impact applications. In this study, we investigate the pressure-dependent dynamic strength behavior of OFHC copper at strain rates of 10^5 s^-1 and pressures approaching 50 GPa. High-pressure shear plate impact experiments are conducted using a sandwich configuration to extract the complete stress-strain response of the materials at high pressures. OFHC copper displays a strong pressure hardening in strength, and the experimental results reveal that the scaling of shear modulus and strength are quite different; they were previously assumed to be the same in material models. The atomistic mechanisms responsible for pressure-dependent strength are explored using molecular dynamics simulations.