We present the first reliable determination of the orbital period of the recurrent nova V2487 Oph (Nova Oph 1998). We derived a value of 0.753 ± 0.016 d (18.1 ± 0.4 h) from the radial velocity curve of the intense He II λ4686 emission line as detected in time-series X-shooter spectra. The orbital period is significantly shorter than earlier claims, but it makes V2487 Oph one of the longest period cataclysmic variables known. The spectrum of V2487 Oph is prolific in broad Balmer absorptions that resemble a white dwarf spectrum. However, we show that they come from the accretion disc viewed at low inclination. Although highly speculative, the analysis of the radial velocity curves provides a binary mass ratio q ≈ 0.16 and a donor star mass M2 ≈ 0.21 M⊙, assuming the reported white dwarf mass M1 = 1.35 M⊙. A subgiant M-type star is tentatively suggested as the donor star. We were lucky to inadvertently take some of the spectra when V2487 Oph was in a flare state. During the flare, we detected high-velocity emission in the Balmer and He II λ4686 lines exceeding -2000 km s-1 at close to orbital phase 0.4. Receding emission up to 1200 km s-1 at about phase 0.3 is also observed. The similarities with the magnetic cataclysmic variables may point to magnetic accretion on to the white dwarf during the repeating flares.