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Free-floating substellar candidates with estimated theoretical masses of as low as ∼5 Jupiter masses have been found in the ∼3Myr old {sigma} 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. We propose to search for objects of yet lower masses in the cluster and determine the shape of the mass function at low mass. Using new- and (re-analysed) published IZJHK_s[3.6]-[8.0]-band data of an area of 840arcmin², we performed a search for LT-type cluster member candidates in the magnitude range J=19.5-21.5mag, based on their expected magnitudes and colours. Besides recovering the T type object SOri 70 and two other known objects, we find three new cluster member candidates, SOri 72-74, with J≃21mag and within 12arcmin of the cluster centre. They have theoretical masses of 4_–2⁺³M_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 SOri 72 is an L/T transition candidate and SOri 73 a T-type candidate, following the expected cluster sequence in the mid-infrared. Finally, the L-type candidate SOri 74 with lower quality photometry is located at 11.8arcsec (∼4250AU) of a stellar member of σ Orionis and could be a companion. After contaminant correction in the area complete to J=21.1mag, we estimate that there remain between zero and two cluster members in the mass interval 6-4M_Jup. We present SOri 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.