Aims: We aim at obtaining near-infrared photometry and deriving the mass, age, temperature, and surface gravity of WISE J085510.74-071442.5 (J0855-0714), which is the coolest object beyond the solar system currently known. Methods: We used publicly available data from the archives of the Hubble Space Telescope (HST) and the Very Large Telescope (VLT) to determine the emission of this source at 1.153 μm (F110W) and 1.575 μm (CH4-off). J0855-0714 was detected at both wavelengths with a signal-to-noise ratio of ≈10 (F110W) and ≈4 (CH4-off) at the peak of the corresponding point-spread-functions. Results: This is the first detection of J0855-0714 in the H-band wavelengths. We measured 26.31 ± 0.10 and 23.22 ± 0.35 mag in F110W and CH4-off (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 mass and brightness) and at ~1 AU (≈1 mag fainter in the F110W filter) are discarded. By combining the new data with published photometry, including non-detections, we build the spectral energy distribution of J0855-0714 from 0.89 through 22.09 μm, and contrast it against current solar-metallicity models of planetary atmospheres. We determine that the best spectral fit yields a temperature of 225-250 K, a bolometric luminosity of log L/L⊙ = -8.57, and a high surface gravity of log g = 5.0 (cm s-2), which suggests an old age although a gravity this high is not fully compatible with evolutionary models. After comparing our data 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 high atmospheric gravities of log g ⪆ 3.5 (cm s-2). If it had the age of the Sun, J0855-0714 would be a ≈5-MJup free-floating planetary-mass object. Conclusions: J0855-0714 meets the mass values previously determined for free-floating planetary-mass objects discovered in star-forming regions and young stellar clusters. Based on extrapolations of the substellar mass functions of young clusters to the field, as many J0855-0714-like objects as M5-L2 stars may be expected to populate the solar neighborhood.