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 IZJHK s [3.6]−[8.0]-band data of an area of 840 arcmin 2 , 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 +3 −2 M Jup 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 M Jup . 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.