For an airborne passive radar with contaminated reference signals, the clutter caused by multipath (MP) signals involved in the reference channel (MP clutter) corrupts the covariance estimation in space-time adaptive processing (STAP). In order to overcome the severe STAP performance degradation caused by impure reference signals and off-grid effects, a novel MP clutter suppression method based on local search is proposed for airborne passive radar. In the proposed method, the global dictionary is constructed based on the sparse measurement model of MP clutter, and the global atoms that are most relevant to the residual are selected. Then, the local dictionary is designed iteratively, and local searches are performed to match real MP clutter points. Finally, the off-grid effects are mitigated, and the MP clutter is suppressed from all matched atoms. A range of simulations is conducted in order to demonstrate the effectiveness of the proposed method.