  We explore a new way of producing the Rashba spin-orbit coupling (SOC) for
ultracold atoms by using a two-component (spinor) atomic Bose-Einstein
condensate (BEC) confined in a bilayer geometry. The SOC of the Rashba type is
created if the atoms pick up a {\pi} phase after completing a cyclic transition
between four combined spin-layer states composed of two spin and two layer
states. The cyclic coupling of the spin-layer states is carried out by
combining an intralayer Raman coupling and an interlayer laser assisted
tunneling. We theoretically determine the ground-state phases of the
spin-orbit-coupled BEC for various strengths of the atom-atom interaction and
the laser-assisted coupling. It is shown that the bilayer scheme provides a
diverse ground-state phase diagram. In an intermediate range of the atom-light
coupling two interlacing lattices of half- skyrmions and half-antiskyrmions are
spontaneously created. In the strong-coupling regime, where the SOC of the
Rashba-type is formed, the ground state represents plane-wave or standing-wave
phases depending on the interaction between the atoms. A variational analysis
is shown to be in a good agreement with the numerical results.
