An anomalous emission component at energies of a few gigaelectronvolts and located towards the inner Galaxy is present in the Fermi-LAT data. At present, the two most promising explanations are the annihilation of dark matter particles or the presence of a large population of unresolved point sources, most probably millisecond pulsars, at the Galactic Centre. Here, we report an analysis of the excess characteristics using almost eight years of Pass 8 ULTRACLEAN Fermi-LAT data with SkyFACT—a tool that combines image reconstruction with template-fitting techniques. We find that an emission profile that traces stellar mass in the boxy and nuclear bulge is preferred over conventional dark matter profiles. A model including the bulge is favoured over a model with dark matter at 16σ.