We use integral-field spectroscopy from the SAMI Galaxy Survey to identify galaxies that show evidence of recent quenching of star formation. The galaxies exhibit strong Balmer absorption in the absence of ongoing star formation in more than 10% of their spectra within the SAMI field of view. These -strong (HDS) galaxies (HDSGs) are rare, making up only ∼2% (25/1220) of galaxies with stellar mass > 10. The HDSGs make up a significant fraction of nonpassive cluster galaxies (15%; 17/115) and a smaller fraction (2.0%; 8/387) of the nonpassive population in low-density environments. The majority (9/17) of cluster HDSGs show evidence of star formation at their centers, with the HDS regions found in the outer parts of the galaxy. Conversely, the HDS signal is more evenly spread across the galaxy for the majority (6/8) of HDSGs in low-density environments and is often associated with emission lines that are not due to star formation. We investigate the location of the HDSGs in the clusters, finding that they are exclusively within 0.6R200 of the cluster center and have a significantly higher velocity dispersion relative to the cluster population. Comparing their distribution in projected phase space to those derived from cosmological simulations indicates that the cluster HDSGs are consistent with an infalling population that has entered the central 0.5r200,3D cluster region within the last ∼1 Gyr. In the eight of nine cluster HDSGs with central star formation, the extent of star formation is consistent with that expected of outside-in quenching by ram pressure stripping. Our results indicate that the cluster HDSGs are currently being quenched by ram pressure stripping on their first passage through the cluster.