We present the first polarimetric space very long baseline interferometry (VLBI) observations of OJ 287, observed with RadioAstron at 22 GHz during a perigee session on 2014 April 4 and five near-in-time snapshots, together with contemporaneous ground VLBI observations at 15, 43, and 86 GHz. Ground-space fringes were obtained up to a projected baseline of 3.9 Earth diameters during the perigee session, and at a record 15.1 Earth diameters during the snapshot sessions, allowing us to image the innermost jet at an angular resolution of ∼50μ as, the highest ever achieved at 22 GHz for OJ 287. Comparison with ground-based VLBI observations reveals a progressive jet bending with increasing angular resolution that agrees with predictions from a supermassive binary black hole model, although other models cannot be ruled out. Spectral analyses suggest that the VLBI core is dominated by the internal energy of the emitting particles during the onset of a multiwavelength flare, while the parsec-scale jet is consistent with being in equipartition between the particles and magnetic field. Estimated minimum brightness temperatures from the visibility amplitudes show a continued rising trend with projected baseline length up to 1013 K, reconciled with the inverse-Compton limit through Doppler boosting for a jet closely oriented to the line of sight. The observed electric vector position angle suggests that the innermost jet has a predominantly toroidal magnetic field, which, together with marginal evidence of a gradient in rotation measure across the jet width, indicates that the VLBI core is threaded by a helical magnetic field, in agreement with jet formation models.