Aims: 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 M⊙. Methods: We performed a deep photometric and astrometric J- and H-band survey covering an area of ~0.8 deg2 in the Pleiades cluster. The images with completeness and limiting magnitudes of J,H ≈ 20.2 and ≈21.5 mag were acquired ~9 yr apart, allowing us to derive proper motions with a typical precision of ±6 mas yr-1. For the cluster distance and age, the survey is sensitive to Pleiades members with masses in the interval ≈0.2-0.008 M⊙. J- and H-band data were complemented with Z, K, and mid-infrared magnitudes up to 4.6 μm coming from the UKIRT Infrared Deep Sky Survey (UKIDSS), the WISE catalog, 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 = 15.5-18.8 mag, and ZUKIDSS - J ≥ 2.3 mag or Z nondetections for J> 18.8 mag. Results: We found a neat sequence of astrometric and photometric Pleiades substellar member candidates with two or more proper motion measurements and with magnitudes and masses in the intervals J = 15.5-21.2 mag and ≈0.072-0.008 M⊙. The faintest objects show very red near- and mid-infrared colors exceeding those of field high-gravity dwarfs by ≥0.5 mag. This agrees with the reported properties of field young L-type dwarfs and giant planets orbiting stars of ages of ~100 Myr. 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 combination of evolutionary models and colors computed from model atmospheres. The astrometric data suggest that Pleiades brown dwarfs have a proper motion dispersion of 6.4-7.5 mas yr-1, 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. The new discoveries may become benchmark objects for interpreting the observations of the emerging young ultracool population and giant planets around stars in the solar neighborhood.