Torques appear between charge carrier spins and local moments in regions of ferromagnetic media where spatial magnetization gradients occur, such as a domain wall, owing to an exchange interaction. This phenomenon has been predicted by different theories and confirmed in a number of experiments on metallic and semiconductor ferromagnets. Understanding the magnitude and orientation of such spin-torques is an important problem for spin-dependent transport and current-driven magnetization dynamics, as domain-wall motion underlies a number of emerging spintronic technologies. One outstanding issue concerns the non-adiabatic spin-torque component β, which has an important role in wall dynamics, but no clear consensus has yet emerged over its origin or magnitude. Here, we report an experimental measurement of β in perpendicularly magnetized films with narrow domain walls (1-10 nm). By studying thermally activated wall depinning, we deduce β from the variation of the Arrhenius transition rate with applied currents. Surprisingly, we find β to be small and relatively insensitive to the wall width, which stands in contrast to predictions from transport theories. In addition, we find β to be close to the Gilbert damping constant α, which, in light of similar results on planar anisotropy systems, suggests a universal origin for the non-adiabatic torque.