  We provide an early assessment of the imaging capabilities of the Euclid
space mission to probe deeply into nearby star-forming regions and associated
very young open clusters, and in particular to check to what extent it can shed
light on the new-born free-floating planet population. This paper focuses on a
low-reddening region observed in just one Euclid pointing where the dust and
gas has been cleared out by the hot sigma Orionis star. One late-M and six
known spectroscopically confirmed L-type substellar members in the sigma
Orionis cluster are 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 allow us to focus on bona-fide point
sources. A cleaned sample of sigma Orionis cluster substellar members has been
produced and the initial mass function (IMF) has been estimated by combining
Euclid and Gaia data. Our sigma Orionis substellar IMF is consistent with a
power-law distribution with no significant steepening at the planetary-mass
end. No evidence of a low-mass cutoff is found down to about 4 Jupiter masses
at the young age (3 Myr) of the sigma Orionis open cluster.
