Airborne radars for maritime surveillance usually operate in scanning mode to detect marine objects over 360 degrees. More recently, the concept of forming inverse synthetic aperture radar (ISAR) images while the antenna is scanning, called circular scan ISAR in this paper, has drawn attention. Circular scan ISAR enables a large coverage area, sequential imaging of multiple targets, and the formation of an image sequence of a target. However, the rapid antenna scanning results in a short accumulation time of the target and reduces the Doppler resolution of the system. This paper analyzes the limit values of the Doppler resolution improvement ratio when super-resolution imaging is performed using only the compressed sensing method by relating the limit values to the restricted isometry constant (RIC) value of the sensing matrices. The limit values are found to be relatively small when the sparsity degree of the sparse signal is relatively large. Therefore, the super-resolution imaging of ships for circular scan ISAR is carried out by extrapolating data in the azimuth direction first and then reconstructing the data via compressed sensing to obtain low side-lope levels. Finally, the constant false alarm rate (CFAR) detection is performed on the matrix composed of calculated sparse coefficients to improve the signal-to-clutter noise ratio (SCNR). The simulation and real-data processing results are provided to verify the effectiveness of the presented algorithm.