  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.
