  We show that the newly discovered BiS$_2$-based superconductors may have a
dominant triplet pairing component, in addition to a subdominant singlet
component arising from the spin-orbital coupling. The pairing respects
time-reversal symmetry. The dominant triplet gap causes gap sign changes
between the spin-split Fermi pockets. Within a pocket, the gap function
respects $d^*_{x^2-y^2}$-wave symmetry, where the star indicates joint
spin-lattice rotations. Below the Lifshitz filling level the gap is nodelss and
the superconducting state is weak topological. Above the Lifshitz points the
gap becomes nodal. The dominant triplet pairing is consistent with the
experimental upper critical field exceeding the Pauli limit.
