Near real-time ship monitoring is crucial for ensuring safety and security at sea. Established ship monitoring systems are the automatic identification system (AIS) and marine radars. However, not all ships are committed to carry an AIS transponder and the marine radars suffer from limited visibility. For these reasons, airborne radars can be used as an additional and supportive sensor for ship monitoring, especially on the open sea. State-of-the-art algorithms for ship detection in radar imagery are based on constant false alarm rate (CFAR). Such algorithms are pixel-based and therefore it can be challenging in practice to achieve near real-time detection. This letter presents two object-oriented ship detectors based on the faster region-based convolutional neural network (R-CNN). The first detector operates in time domain and the second detector operates in Doppler domain of airborne Range-Compressed (RC) radar data patches. The Faster R-CNN models are trained on thousands of real X-band airborne RC radar data patches containing several ship signals. The robustness of the proposed object-oriented ship detectors is tested on multiple scenarios, showing high recall performance of the models even in very dense multitarget scenarios in the complex inshore environment of the North Sea.