Aims:We investigate the mass function in the substellar domain down to a few Jupiter masses in the young σ Orionis open cluster (3±2 Ma, d = 360^+70_-60 pc). Methods: We have performed a deep IJ-band search, covering an area of 790 arcmin2 close to the cluster centre. This survey was complemented with an infrared follow-up in the HK_s- and Spitzer 3.6-8.0 μm-bands. Using colour-magnitude diagrams, we have selected 49 candidate cluster members in the magnitude interval 16.1 mag < I < 23.0 mag. Results: Accounting for flux excesses at 8.0 μm and previously known spectral features of youth, we identify 30 objects as bona fide cluster members. Four are first identified from our optical-near infrared data. Eleven have most probable masses below the deuterium burning limit which we therefore classify as candidate planetary-mass objects. The slope of the substellar mass spectrum (Δ N / Δ {M} ≈ a { M}-α) in the mass interval 0.11 M_&sun; < {M} < 0.006 M_&sun; is α = +0.6 ± 0.2. Any mass limit to formation via opacity-limited fragmentation must lie below 0.006 M_&sun;. The frequency of σ Orionis brown dwarfs with circumsubstellar discs is 47±9 %. Conclusions: The continuity in the mass function and in the frequency of discs suggests that very low-mass stars and substellar objects, even below the deuterium-burning mass limit, share the same formation mechanism. Alexander von Humboldt Fellow at the Max-Planck-Institut für Astronomie Appendices are only available in electronic form at http://www.aanda.org