OGLE-2014-BLG-1186: gravitational microlensing providing evidence for a planet orbiting the foreground star or for a close binary source?

(abridged) Using the particularly long gravitational microlensing event OGLE-2014-BLG-1186 with a time-scale $t_\mathrm{E}$ ~ 300 d, we present a methodology for identifying the nature of localised deviations from single-lens point-source light curves, which ensures that 1) the claimed signal is substantially above the noise floor, 2) the inferred properties are robustly determined and their estimation not subject to confusion with systematic noise in the photometry, 3) there are no alternative viable solutions within the model framework that might have been missed. Annual parallax and binarity could be separated and robustly measured from the wing and the peak data, respectively. We find matching model light curves that involve either a binary lens or a binary source. Our binary-lens models indicate a planet of mass $M_2$ = (45 $\pm$ 9) $M_\oplus$, orbiting a star of mass $M_1$ = (0.35 $\pm$ 0.06) $M_\odot$, located at a distance $D_\mathrm{L}$ = (1.7 $\pm$ 0.3) kpc from Earth, whereas our binary-source models suggest a brown-dwarf lens of $M$ = (0.046 $\pm$ 0.007) $M_\odot$, located at a distance $D_\mathrm{L}$ = (5.7 $\pm$ 0.9) kpc, with the source potentially being a (partially) eclipsing binary involving stars predicted to be of similar colour given the ratios between the luminosities and radii. The ambiguity in the interpretation would be resolved in favour of a lens binary by observing the luminous lens star separating from the source at the predicted proper motion of $\mu$ = (1.6 $\pm$ 0.3) mas yr$^{-1}$, whereas it would be resolved in favour of a source binary if the source could be shown to be a (partially) eclipsing binary matching the obtained model parameters. We experienced that close binary source stars pose a challenge for claiming the detection of planets by microlensing in events where the source passes very close to the lens star hosting the planet.