We report the observation of extraordinarily robust zero-resistance superconductivity in the pressurized (TaNb)_0.67(HfZrTi)_0.33 high-entropy alloy--a material with a body-centered-cubic crystal structure made from five randomly distributed transition-metal elements. The transition to superconductivity (T_c) increases from an initial temperature of 7.7 K at ambient pressure to 10 K at ∼60 GPa, and then slowly decreases to 9 K by 190.6 GPa, a pressure that falls within that of the outer core of the earth. We infer that the continuous existence of the zero-resistance superconductivity from 1 atm up to such a high pressure requires a special combination of electronic and mechanical characteristics. This high-entropy alloy superconductor thus may have a bright future for applications under extreme conditions, and also poses a challenge for understanding the underlying quantum physics.