Context: The knowledge of the initial mass function (IMF) in open clusters constitutes one way of constraining the formation of low-mass stars and brown dwarfs, along with the frequency of multiple systems and the properties of disks. Aims: The aim of the project is to determine the shape of the mass function in the low-mass and substellar regimes in the σ Orionis cluster (~3 Myr, ~352 pc, solar metallicity) as accurately as possible and compare it with the results in other clusters. Methods: We have analysed the near-infrared photometric data from the fourth data release (DR4) of the UKIRT Infrared Deep Sky Suvey (UKIDSS) Galactic clusters survey (GCS) to derive the cluster luminosity and mass functions, evaluate the extent of the cluster, and study the distribution and variability of low-mass stars and brown dwarfs down to the deuterium-burning limit. Results: We have recovered most of the previously published members and found a total of 287 candidate members within the central 30 arcmin in the 0.5-0.009 M&sun; mass range, including new objects not previously reported in the literature. This new catalogue represents a homogeneous dataset of brown dwarf member candidates over the central 30 arcmin of the cluster. The expected photometric contamination by field objects with similar magnitudes and colours to σ Orionis members is ~15%. We present evidence of variability at the 99.5% confidence level over ~yearly timescales in 10 member candidates that exhibit signs of youth and the presence of disks. The level of variability is low (≤0.3 mag) and does not impact the derivation of the cluster luminosity and mass functions. Furthermore, we find a possible dearth of brown dwarfs within the central five arcmin of the cluster, which is not caused by a lower level of photometric sensitivity around the massive, O-type multiple star σ Ori in the GCS survey. Using state-of-the-art theoretical models, we derived the luminosity and mass functions within the central 30 arcmin from the cluster centre, with completeness down to J = 19 mag, corresponding to masses ranging from 0.5 M&sun; down to the deuterium-burning mass boundary (~0.013 M&sun;). Conclusions: The mass function of σ Orionis in this mass interval shows a power law index α = 0.5±0.2 (when expressed as dN/dM ∝ M-α), which agrees with the one derived for the 3-5 Myr cluster Upper Sco (based on similar data obtained with the GCS) in the same mass range. This work is based in part on data obtained as part of the UKIRT Infrared Deep Sky Survey. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the UK. Appendices are only available at http://www.aanda.org