We report on our analysis of the VISTA Orion ZY JHKs photometric data (completeness magnitudes of Z = 22.6 and J = 21.0 mag) focusing on a circular area of 2798.4 arcmin2 around the young σ Orionis star cluster (~3 Myr, ~352 pc, and solar metallicity). The combination of the VISTA photometry with optical, WISE and Spitzer data allows us to identify a total of 210 σ Orionis member candidates with masses in the interval 0.25-0.004 M &sun;, 23 of which are new planetary-mass object findings. These discoveries double the number of cluster planetary-mass candidates known so far. One object has colors compatible with a T spectral type. The σ Orionis cluster harbors about as many brown dwarfs (69, 0.072-0.012 M &sun;) and planetary-mass objects (37, 0.012-0.004 M &sun;) as very low mass stars (104, 0.25-0.072 M &sun;). Based on Spitzer data, we derive a disk frequency of ~40% for very low mass stars, brown dwarfs, and planetary-mass objects in σ Orionis. The radial density distributions of these three mass intervals are alike: all are spatially concentrated within an effective radius of 12' (1.2 pc) around the multiple star σ Ori, and no obvious segregation between disk-bearing and diskless objects is observed. Using the VISTA data and the Mayrit catalog, we derive the cluster mass spectrum (ΔN/ΔM ~ M -α) from ~19 to 0.006 M &sun; (VISTA ZJ completeness), which is reasonably described by two power-law expressions with indices of α = 1.7 ± 0.2 for M > 0.35 M &sun;, and α = 0.6 ± 0.2 for M < 0.35 M &sun;. The σ Orionis mass spectrum smoothly extends into the planetary-mass regime down to 0.004 M &sun;. Our findings of T-type sources (<0.004 M &sun;) in the VISTA σ Orionis exploration appear to be smaller than what is predicted by the extrapolation of the cluster mass spectrum down to the survey J-band completeness.