(Abridged) We aim at identifying the least massive population of the solar metallicity, young (120 Myr), nearby (133.5 pc) Pleiades star cluster with the ultimate goal of understanding the physical properties of intermediate-age, free-floating, low-mass brown dwarfs and giant planetary-mass objects, and deriving the cluster substellar mass function across the deuterium-burning mass limit at ~0.012 Msol. We performed a deep photometric and astrometric J- and H-band survey covering an area of ~0.8 deg^2. The images with completeness and limiting magnitudes of J,H ~ 20.2 and ~ 21.5 mag were acquired ~9 yr apart (proper motion precision of +/-6 mas/yr). J- and H-band data were complemented with Z, K, and mid-infrared magnitudes up to 4.6 micron coming from UKIDSS, WISE, and follow-up observations of our own. Pleiades member candidates were selected to have proper motions compatible with that of the cluster, and colors following the known Pleiades sequence in the interval J = 15.5-8.8 mag, and Z_UKIDSS - J > 2.3 mag or Z nondetections for J > 18.8 mag. We found a neat sequence of astrometric and photometric Pleiades substellar member candidates in the intervals J = 15.5-21.2 mag and ~0.072-0.008 Msol. The faintest objects show very red near- and mid-infrared colors exceeding those of field high-gravity dwarfs by >0.5 mag. The Pleiades photometric sequence does not show any color turn-over because of the presence of photospheric methane absorption down to J = 20.3 mag, which is about 1 mag fainter than predicted by the color-computed models. Pleiades brown dwarfs have a proper motion dispersion of 6.4-7.5 mas/yr and are dynamically relaxed at the age of the cluster. The Pleiades mass function extends down to the deuterium burning-mass threshold, with a slope fairly similar to that of other young star clusters and stellar associations.
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