We report the discovery of spatially extended, non‐thermal‐like emission components in Chandra X‐ray spectra for five of a sample of seven massive, merging galaxy clusters with powerful radio haloes. The emission components can be fitted by power‐law models with mean photon indices in the range 1.5 < Γ < 2.0. A control sample of regular, dynamically relaxed clusters, without radio haloes but with comparable mean thermal temperatures and luminosities, shows no compelling evidence for similar components. Detailed X‐ray spectral mapping reveals the complex thermodynamic states of the radio halo clusters. Our deepest observations, of the Bullet Cluster 1E 0657−56, demonstrate a spatial correlation between the strongest power‐law X‐ray emission, highest thermal pressure and brightest 1.34 GHz radio halo emission in this cluster. We confirm the presence of a shock front in the 1E 0657−56 and report the discovery of a new, large‐scale shock front in Abell 2219. We explore possible origins for the power‐law X‐ray components. These include inverse‐Compton scattering of cosmic microwave background photons by relativistic electrons in the clusters; bremsstrahlung from suprathermal electrons energized by Coulomb collisions with an energetic, non‐thermal proton population; and synchrotron emission associated with ultrarelativistic electrons. Interestingly, we show that the power‐law signatures may also be due to complex temperature and/or metallicity structure in clusters particularly in the presence of metallicity gradients. In this case, an important distinguishing characteristic between the radio halo clusters and control sample of predominantly cool‐core clusters is the relatively low central X‐ray surface brightness of the former. Our results have implications for previous discussions of soft excess X‐ray emission from clusters and highlight the importance of further deep X‐ray and radio mapping, coupled with new hard X‐ray, γ‐ray and TeV observations, for improving our understanding of the non‐thermal particle populations in these systems.