Context. The stellar population of the 30 Doradus star-forming region in the Large Magellanic Cloud contains a subset of apparently single, rapidly rotating O-type stars. The physical processes leading to the formation of this cohort are currently uncertain. Aims. One member of this group, the late O-type star VFTS 399, is found to be unexpectedly X-ray bright for its bolometric luminosity − in this study we aim to determine its physical nature and the cause of this behaviour. Methods. To accomplish this we performed a time-resolved analysis of optical, infrared and X-ray observations. Results. We found VFTS 399 to be an aperiodic photometric variable with an apparent near-IR excess. Its optical spectrum demonstrates complex emission profiles in the lower Balmer series and select He i lines − taken together these suggest an OeBe classification.The highly variable X-ray luminosity is too great to be produced by a single star, while the hard, non-thermal nature suggests the presence of an accreting relativistic companion. Finally, the detection of periodic modulation of the X-ray lightcurve is most naturally explained under the assumption that the accretor is a neutron star. Conclusions. VFTS 399 appears to be the first high-mass X-ray binary identified within 30 Dor, sharing many observational characteristics with classical Be X-ray binaries. Comparison of the current properties of VFTS 399 to binary-evolution models suggests a progenitor mass 25 M for the putative neutron star, which may host a magnetic field comparable in strength to those of magnetars. VFTS 399 is now the second member of the cohort of rapidly rotating "single" O-type stars in 30 Dor to show evidence of binary interaction resulting in spin-up, suggesting that this may be a viable evolutionary pathway for the formation of a subset of this stellar population.