The antiferromagnetic semi-Heusler compound CuMnSb has been investigated under high pressure by electrical resistivity and angle dispersive synchrotron x-ray diffraction measurements to 53 and 36 GPa, respectively. The Néel temperature at ∼50 K is found to initially increase rapidly with pressure, reaching 83 K at 7 GPa. However, near 8 GPa at ambient temperature a sluggish first-order structural transition begins from a semimetallic cubic phase to a likely semimetallic tetragonal phase; thermal cycling to 355°C at 9.6 GPa serves to complete the transition. In the tetragonal phase no sign of magnetic ordering is visible in the resistivity R(T) over the measured temperature range 4-295 K. This suggests that magnetic ordering may have shifted to temperatures well above ambient. Indeed, density functional calculations find the magnetic ground state in the tetragonal phase to be antiferromagnetic. Following decompression to 1 bar at ambient temperature, the high-pressure tetragonal phase is retained.