  We address a possibility of creating soliton states in oblate
binary-fermionic clouds in the framework of the density-functional theory,
which includes the spin-orbit coupling (SOC) and nonlinear attraction between
spin-up and down-polarized components of the spinor wave function. In the limit
when the inter-component attraction is much stronger than the effective
intra-component Pauli repulsion, the resulting model also represents a system
of Gross-Pitaevskii equations for a binary Bose-Einstein condensate including
the SOC effect. We show that the model gives rise to two-dimensional quiescent
composite solitons in free space. A stability region is identified for solitons
of the mixed-mode type (which feature mixtures of zero-vorticity and vortical
terms in both components), while solitons of the other type, semi-vortices
(with the vorticity carried by one component) are unstable. Due to breaking of
the Galilean invariance by SOC, the system supports moving solitons with
velocities up to a specific critical value. Collisions between moving solitons
are briefly considered too. The collisions lead, in particular, to a
quasi-elastic rebound, or an inelastic outcome, which features partial merger
of the solitons.
