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Cooling the trapped atoms toward their motional ground states is key to applications of quantum simulation and quantum computation. By utilizing nonreciprocal couplings between constituent atoms, we present an intriguing dark-state cooling scheme in $Λ$-type three-level structure, which is shown superior than the conventional electromagnetically-induced-transparency cooling in a single atom. The effective nonreciprocal couplings can be facilitated either by an atom-waveguide interface or a free-space photonic quantum link. By tailoring system parameters allowed in dark-state cooling, we identify the parameter regions of better cooling performance with an enhanced cooling rate. We further demonstrate a mapping to the dark-state sideband cooling under asymmetric laser driving fields, which shows a distinct heat transfer and promises an outperforming dark-state sideband cooling assisted by collective spin-exchange interactions.