Emission from active galactic nuclei (AGNs) and from neutral pion decay are the two most natural mechanisms that could establish a galaxy cluster as a source of gamma rays in the GeV regime. We revisit this problem by using 52.5 months of Fermi-LAT data above 10 GeV and stacking 55 clusters from the HIFLUCGS sample of the X-ray brightest clusters. The choice of >10 GeV photons is optimal from the point of view of angular resolution, while the sample selection optimizes the chances of detecting signatures of neutral pion decay, arising from hadronic interactions of relativistic protons with an intracluster medium, which scale with the X-ray flux. In the stacked data we detected a signal for the central 0.25 deg circle at the level of 4.3 sigma. Evidence for a spatial extent of the signal is marginal. A subsample of cool-core clusters has a higher count rate of 1.9 +/- 0.3 per cluster compared to the subsample of non-cool core clusters at 1.3 +/- 0.2. Several independent arguments suggest that the contribution of AGNs to the observed signal is substantial, if not dominant. No strong support for the large contribution of pion decay was found. In terms of a limit on the relativistic proton energy density, we derived an upper limit of similar or equal to 2% relative to the gas thermal energy density, provided that the spectrum of relativistic protons is hard (s = 4.1 in dN alpha p(-s) d(3) p). This estimate assumes that relativistic and thermal components are mixed. For softer spectra the upper limits are weaker and equal to 3% for s = 4.2. 4% for s = 4.3, and 6% for s = 4.4.