Context: Free-floating substellar candidates with estimated theoretical masses of as low as ∼5 Jupiter masses have been found in the ∼3 Myr old σ Orionis open cluster. As the overlap with the planetary mass domain increases, the question of how these objects form becomes important. The determination of their number density and whether a mass cut-off limit exists is crucial to understanding their formation. Aims: We propose to search for objects of yet lower masses in the cluster and determine the shape of the mass function at low mass. Methods: Using new- and (re-analysed) published IZJHKs[3.6]-[8.0]-band data of an area of 840 arcmin2, we performed a search for LT-type cluster member candidates in the magnitude range J=19.5-21.5 mag, based on their expected magnitudes and colours. Results: Besides recovering the T type object S Ori 70 and two other known objects, we find three new cluster member candidates, S Ori 72-74, with J≈21 mag and within 12 arcmin of the cluster centre. They have theoretical masses of 4-2 +3 MJup and are among the least massive free-floating objects detected by direct imaging outside the Solar System. The photometry in archival Spitzer [3.6]-[5.8]-band images infers that S Ori 72 is an L/T transition candidate and S Ori 73 a T-type candidate, following the expected cluster sequence in the mid-infrared. Finally, the L-type candidate S Ori 74 with lower quality photometry is located at 11.8 arcsec (∼4250 AU) of a stellar member of σ Orionis and could be a companion. After contaminant correction in the area complete to J=21.1 mag, we estimate that there remain between zero and two cluster members in the mass interval 6-4 MJup. Conclusions: We present S Ori 73, a new candidate T type and candidate σ Orionis member of a few Jupiter masses. Our result suggests a possible turnover in the substellar mass spectrum below ∼6 Jupiter masses, which could be investigated further by wider and deeper photometric surveys.