  Coy Dark Matter removes the tension between the traditional WIMP paradigm of
Dark Matter and the latest exclusion bounds from direct detection experiments.
In this paper we present a leptophilic Coy Dark Matter model that, on top of
explaining the spatially extended 1-5 GeV $\gamma$-ray excess detected at the
Galactic Center, reconciles the measured anomalous magnetic moment of muon with
the corresponding Standard Model prediction. The annihilation channel of DM is
$\chi\chi \to \tau\bar\tau$ with the DM mass $m_\chi =
9.43\,(^{+.063}_{-0.52}\,{\rm stat.})\,(\pm 1.2 \,{\rm sys.})$ GeV given by
best-fit of the $\gamma$-ray excess. Fitting the measured anomalous magnetic
moment of the muon requires instead a pseudoscalar mediator with a minimal mass
$m_a = 12^{+7}_{-3}$ GeV.
