We investigate the buckling dynamics of an elastic filament impacted axially by a falling liquid droplet, and identify the buckling modes through a combination of experimental and theoretical analyses. A phase diagram is constructed on a plane defined by two primary parameters: the falling height and the filament length. Two transition boundaries are observed, with one marking the occurrence of dynamic buckling and the other separating the buckling regime into two distinct modes. Notably, the hydrodynamic viscous force of the liquid dominates during the impact, with the dynamic buckling instability predicted by a single elastoviscous number. The critical load is twice the critical static load, which is, however, lower for the deformable droplet utilized in our study, as compared to a solid object. An additional time-dependent simulation on a longer filament exhibits a higher buckling mode, succeeded by a more distinct coarsening process than our experimental observations.