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 multiwavelength studies of the source field. Its gamma-ray emission can be described with a power law (Gamma = 2.0 +/- 0.2) with an exponential cut-off (E-c = 2.9 +/- 1.6 GeV), while the flux shows significant long-term variations. From analysis of archival Neil Gehrels Swift Observatory X-Ray Telescope data, we find only one X-ray source in the LAT's 2 sigma error region. Within a 3.7 arcsec radius error circle of the X-ray source, there is only one optical object down to r' similar to 23 mag. Time-resolved photometry of the optical object indicates a likely 2.5 h 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 subluminous disc state, although the other possibility, the binary in a rotation powered state showing the optical emission lines due to intrabinary interaction processes, cannot be excluded. Further observational studies will help to determine detailed properties of this candidate MSP binary and thus clarify its current state.