  We investigate the pairing physics in a three-component Fermi-Fermi mixture,
where a few impurities are immersed in a non-interacting spin-$\frac{1}{2}$
Fermi gas with synthetic spin-orbit coupling (SOC), and interact attractively
with one spin species in the Fermi gas. Due to the interplay of SOC and
spin-selective interaction, the molecular state intrinsically acquires a
non-zero center-of-mass momentum, which results in a new type of Fulde-Ferrell
(FF) pairing in spin-orbit coupled Fermi systems. The existence of the Fermi
sea can also lead to the competition between FF-like molecular states with
different center-of-mass momenta, which corresponds to a first-order transition
between FF phases in the thermodynamic limit. As the interaction strength is
tuned, a polaron-molecule transition occurs in the highly imbalanced system,
where the boundary varies non-monotonically with SOC parameters and gives rise
to the reentrance of polaron states. The rich physics in this system can be
probed using existing experimental techniques.
