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.
