We provide an early assessment of the imaging capabilities of the Euclid space mission to deeply probe nearby star-forming regions and associated very young open clusters, and in particular, to determine to which extent it can shed light into the newborn free- floating planet population. This paper focusses on a low-reddening region observed in just one Euclid pointing. The dust and gas has been cleared out from the region by the hot σ Orionis star. One late-M and six known spectroscopically confirmed L-type ultracool members in the σ Orionis cluster were used as benchmarks to provide a high-purity procedure to select new candidate members with Euclid. The exquisite angular resolution and depth delivered by the Euclid instruments allowed us to focus on bona fide point sources. A cleaned sample of σ Orionis cluster substellar members was produced, and the initial mass function (IMF) was estimated by combining Euclid and Gaia data. Our σ Orionis substellar IMF is consistent with a power-law distribution without a significant steepening at the planetary-mass end. No evidence of a low-mass cutoff is found down to the detection limit of this study at 4 Jupiter masses in the very young σ Orionis open cluster.