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We investigate families of soliton solutions in a spin-orbit coupledBose-Einstein condensate embedded in an optical lattice, which bifurcate fromthe nearly flat lowest band. Unlike the conventional gap solitons the obtainedsolutions have the shape well approximated by a Wannier function (or a fewWannier functions) of the underlying linear Hamiltonian with amplitudes varyingalong the family and with nearly constant widths. The Wannier solitons (WSs)sharing all symmetries of the system Hamiltonian are found to be stable. Suchsolutions allow for the construction of Wannier breathers, that can be viewedas nonlinearly coupled one-hump solitons. The breathers are well described by afew-mode model and manifest stable behavior either in an oscillatory regimewith balanced average populations or in a self-trapping regime characterized byunbalanced atomic populations of the local potential minima (similarly to theconventional boson Josephson junction), with the frequencies controlled by theinter-atomic interactions.

Wannier solitons in spin-orbit-coupled Bose-Einstein condensates in optical lattices with a flat-band