Current microlensing surveys are sensitive to free-floating planets down to Earth-mass objects. All published microlensing events attributed to unbound planets were identified based on their short timescale (below 2 d), but lacked an angular Einstein radius measurement (and hence lacked a significant constraint on the lens mass). Here we present the discovery of a Neptune-mass free-floating planet candidate in the ultrashort (tE=0.320±0.003 d) microlensing event OGLE-2016-BLG-1540. The event exhibited strong finite source effects, which allowed us to measure its angular Einstein radius of θE=9.2±0.5 uas. There remains, however, a degeneracy between the lens mass and distance. The combination of the source proper motion and source-lens relative proper motion measurements favors a Neptune-mass lens located in the Galactic disk. However, we cannot rule out that the lens is a Saturn-mass object belonging to the bulge population. We exclude stellar companions up to 15 au. Owing to the relatively large relative lens-source proper motion, any stellar companions should be detectable using the high-resolution imaging in the relatively near future.