  Pulsar timing arrays (PTAs) can detect disturbances in the fabric of
spacetime on a galactic scale by monitoring the arrival time of pulses from
millisecond pulsars (MSPs). Recent advancements have enabled the use of
$\gamma$-ray radiation emitted by MSPs, in addition to radio waves, for PTA
experiments. Wave dark matter (DM), a prominent class of DM candidates, can be
detected with PTAs due to its periodic perturbations of the spacetime metric.
In response to this development, we perform in this Letter a first analysis of
applying the $\gamma$-ray PTA to detect the ultralight axion-like wave DM, with
the data of Fermi Large Area Telescope (Fermi-LAT). Despite its much smaller
collecting area, the Fermi-LAT $\gamma$-ray PTA demonstrates a promising
sensitivity potential. We show that the upper limits not far from those of the
dedicated radio-PTA projects can be achieved. Moreover, we initiate a
cross-correlation analysis using the data of two Fermi-LAT pulsars. The
cross-correlation of phases, while carrying key information on the source of
the spacetime perturbations, has been ignored in the existing data analyses for
the wave DM detection with PTAs. Our analysis indicates that taking this
information into account can improve the sensitivity to wave DM by $\gtrsim
50\%$ at masses below $10^{-23}$ eV.
