  The possible gamma-ray excess in the inner Galaxy and the Galactic center
(GC) suggested by Fermi-LAT observations has triggered a large number of
studies. It has been interpreted as a variety of different phenomena such as a
signal from WIMP dark matter annihilation, gamma-ray emission from a population
of millisecond pulsars, or emission from cosmic rays injected in a sequence of
burst-like events or continuously at the GC. We present the first comprehensive
study of model systematics coming from the Galactic diffuse emission in the
inner part of our Galaxy and their impact on the inferred properties of the
excess emission at Galactic latitudes $2^\circ<|b|<20^\circ$ and 300 MeV to 500
GeV. We study both theoretical and empirical model systematics, which we deduce
from a large range of Galactic diffuse emission models and a principal
component analysis of residuals in numerous test regions along the Galactic
plane. We show that the hypothesis of an extended spherical excess emission
with a uniform energy spectrum is compatible with the Fermi-LAT data in our
region of interest at $95\%$ CL. Assuming that this excess is the extended
counterpart of the one seen in the inner few degrees of the Galaxy, we derive a
lower limit of $10.0^\circ$ ($95\%$ CL) on its extension away from the GC. We
show that, in light of the large correlated uncertainties that affect the
subtraction of the Galactic diffuse emission in the relevant regions, the
energy spectrum of the excess is equally compatible with both a simple broken
power-law of break energy $2.1\pm0.2$ GeV, and with spectra predicted by the
self-annihilation of dark matter, implying in the case of $\bar{b}b$ final
states a dark matter mass of $49^{+6.4}_{-5.4}$ GeV.
