The white dwarf SDSS J124043.01+671034.68 (SDSS J1240+6710) was previously found to have an oxygen-dominated atmosphere with significant traces of neon, magnesium, and silicon. A possible origin via a violent late thermal pulse or binary interactions has been suggested to explain this very unusual photospheric composition. We report the additional detection of carbon, sodium, and aluminium in far-ultraviolet and optical follow-up spectroscopy. No iron-group elements are detected, with tight upper limits on titanium, iron, cobalt, and nickel, suggesting that the star underwent partial oxygen burning, but failed to ignite silicon burning. Modelling the spectral energy distribution and adopting the distance based on the Gaia parallax, we infer a low white dwarf mass, M-wd = 0.41 +/- 0.05 M-circle dot. The large space velocity of SDSS J1240+6710, computed from the Gaia proper motion and its radial velocity, is compatible with a Galactic rest-frame velocity of similar or equal to 250 km s(-1) in the opposite direction with respect to the Galactic rotation, strongly supporting a binary origin of this star. We discuss the properties of SDSS J1240+6710 in the context of the recently identified survivors of thermonuclear supernovae, the D-6 and LP 40-365 stars, and conclude that it is unlikely related to either of those two groups. We tentatively suggest that SDSS J1240+6710 is the partially burned remnant of a low-mass white dwarf that underwent a thermonuclear event.