(Abridged) We aim at measuring the near-infrared photometry, and deriving the mass, age, temperature, and surface gravity of WISE J085510.74-071442.5 (J0855-0714), which is the coolest known object beyond the Solar System as of today. We use publicly available data from the archives of the HST and the VLT to determine the emission of this source at 1.153 micron (F110W) and 1.575 micron (CH_4). J0855-0714 is detected at both wavelengths with signal-to-noise ratio of ~10 (F110W) and ~4 (CH_4-off) at the peak of the corresponding PSFs. This is the first detection of J0855-0714 in the H-band. We measure 26.31 +/- 0.10 and 23.22 +/- 0.35 mag in F110W and CH_4 (Vega system). J0855-0714 remains unresolved in the HST images that have a spatial resolution of 0.22. Companions at separations of 0.5 AU (similar brightness) and at ~1 AU (~1 mag fainter in the F110W filter) are discarded. By combining the new data with published photometry, we build the spectral energy distribution of J0855-0714 from 0.89 to 22.09 micron, and contrast it against state-of-the-art solar-metallicity models of planetary atmospheres. We determine a temperature of 225-250 K, a bolometric luminosity of log L/Lsol = -8.57, and a high surface gravity of log g = 5.0 (cm/s2), which suggests an old age although such a high gravity is not fully compatible with evolutionary models. After comparison with the cooling theory for brown dwarfs and planets, we infer a mass in the interval 2-10 Mjup for ages of 1-12 Gyr and log g > 3.5 (cm/s2). At the age of the Sun, J0855-0714 would be a ~5-Mjup free-floating planetary-mass object. J0855-0714 may represent the old image of the free-floating planetary-mass objects of similar mass discovered in star-forming regions and young stellar clusters. As many J0855-0714-like objects as M5-L2 stars may be expected to populate the solar neighborhood.
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