  4FGL J0935.3+0901 is a gamma-ray source detected by the Large Area Telescope
(LAT) onboard the Fermi Gamma-Ray Space Telescope. We have conducted detailed
analysis of the LAT data for this source and multi-wavelength studies of the
source field. Its gamma-ray emission can be described with a power law
($\Gamma=2.0\pm0.2$) with an exponential cutoff ($E_c=2.9\pm1.6$ GeV), while
the flux shows significant long-term variations. From analysis of archival Neil
Gehrels Swift X-ray Telescope data, we find only one X-ray source in the LAT's
2$\sigma$ error region. Within a 3.7" radius error circle of the X-ray source,
there is only one optical object down to $r'\sim$23 mag. Time-resolved
photometry of the optical object indicates a likely 2.5~hr periodic modulation,
while its spectrum shows double-peaked hydrogen and helium emission lines
(similar to those seen in accretion discs in low-mass X-ray binaries).
Combining these results, we conclude that we have discovered a compact X-ray
emitting binary in likely association with 4FGL J0935.3+0901, i.e., a
millisecond pulsar (MSP) binary. We discuss the implication of the optical
spectral features: this binary could be a transitional MSP system at a
sub-luminous disc state, although the other possibility, the binary in a
rotation-powered state showing the optical emission lines due to intrabinary
interaction processes, can not be excluded. Further observational studies will
help determine detailed properties of this candidate MSP binary and thus
clarify its current state.
