Aims:We searched for infrared flux excesses of planetary-mass candidates in the σ Orionis cluster (~3 Myr, ~350 pc). Methods: Using IJHKs data from the literature and the [3.6], [4.5], [5.8], and [8.0] IRAC images of the σ Orionis cluster from the Spitzer Space Telescope public archives, we constructed colour-colour diagrams and spectral energy distributions from 0.8 to 8.0 μm of cluster candidates fainter than J = 18.0 mag, i.e. the planetary-mass borderline for σ Orionis. Results: Infrared flux excesses are detected longward of 5 μm in seven objects (S Ori 54, 55, 56, 58, 60, S Ori J053956.8-025315 and S Ori J053858.6-025228) with masses estimated in the range 7-14 M_Jup. Emission at shorter wavelengths (4.5 μm) in excess of the photosphere is probably observed in S Ori 56 and S Ori J053858.6-025228. The faintest and least massive object, S Ori 60, exhibits flux excess only at 8 μm. We ascribe these infrared excesses to the presence of circumsubstellar warm discs, providing additional confirmation for the objects' membership of σ Orionis. The observed incidence of inner discs around planetary-mass objects is ≥50%, which is consistent with the measured inner disc frequency among cluster brown dwarfs and low-mass stars, suggesting that these objects share a common origin. However, there is a trend for the inner disc rate to increase with decreasing mass (from 10 M&sun; through the substellar domain), which may be due to a mass-dependent timescale for the dissipation of the interior discs.