  The HAWC Collaboration has discovered a $\gamma$-ray emission extended about
2 degrees around the Geminga and Monogem pulsar wind nebulae (PWNe) at
$\gamma$-ray energies $E_\gamma >5$ TeV. We analyze, for the first time, almost
10 years of $\gamma$-ray data obtained with the Fermi Large Area Telescope at
$E_\gamma >$ 8 GeV in the direction of Geminga and Monogem. Since these two
pulsars are close the Galactic plane we run our analysis with 10 different
interstellar emission models (IEMs) to study the systematics due to the
modeling of this component. We detect a $\gamma$-ray halo around Geminga with a
significance in the range $7.8-11.8\sigma$ depending on the IEM considered.
This measurement is compatible with $e^+$ and $e^-$ emitted by the PWN, which
inverse-Compton scatter (ICS) with photon fields located within a distance of
about 100 pc from the pulsar, where the diffusion coefficient is estimated to
be around $1.1 \times 10^{27}$ cm$^2$/s at 100 GeV. We include in our analysis
the proper motion of the Geminga pulsar which is relevant for $\gamma$ rays
produced for ICS in the Fermi-LAT energy range. We find that an efficiency of
about $1\%$ for the conversion of the spin-down energy of the pulsar into $e^+$
and $e^-$ is required to be consistent with $\gamma$-ray data from Fermi-LAT
and HAWC. The inferred contribution of Geminga to the $e^+$ flux is at most
$20\%$ at the highest energy AMS-02 data. Our results are compatible with the
interpretation that the cumulative emission from Galactic pulsars explains the
positron excess.
