The search for diffuse non-thermal, inverse Compton (IC) emission from galaxy clusters at hard X-ray energies has been underway for many years, with most detections being either of low significance or controversial. In this work, we investigate 14-195 keV spectra from the Swift Burst Alert Telescope (BAT) all-sky survey for evidence of non-thermal excess emission above the exponentially decreasing tail of thermal emission in the flux-limited HIFLUGCS sample. To account for the thermal contribution at BAT energies, XMM-Newton EPIC spectra are extracted from coincident spatial regions so that both thermal and non-thermal spectral components can be determined simultaneously. We find marginally significant IC components in six clusters, though after closer inspection and consideration of systematic errors we are unable to claim a clear detection in any of them. The spectra of all clusters are also summed to enhance a cumulative non-thermal signal not quite detectable in individual clusters. After constructing a model based on single-temperature fits to the XMM-Newton data alone, we see no significant excess emission above that predicted by the thermal model determined at soft energies. This result also holds for the summed spectra of various subgroups, except for the subsample of clusters with diffuse radio emission. For clusters hosting a diffuse radio halo, a relic, or a mini-halo, non-thermal emission is initially detected at the sim5-sigma confidence level - driven by clusters with mini-halos - but modeling and systematic uncertainties ultimately degrade this significance. In individual clusters, the non-thermal pressure of relativistic electrons is limited to sim10% of the thermal electron pressure, with stricter limits for the more massive clusters, indicating that these electrons are likely not dynamically important in the central regions of clusters.
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