Rapid analysis of satellite imagery within minutes-to-hours of acquisition is increasingly vital for many remote sensing applications, and is an essential component for developing next-generation autonomous and distributed satellite systems. On-satellite machine learning (ML) has the potential for such rapid analysis, by overcoming latency associated with intermittent satellite connectivity to ground stations or relay satellites, but state-of-the-art models are often too large or power-hungry for on-board deployment. Vessel detection using Synthetic Aperture Radar (SAR) is a critical time-sensitive application in maritime security that exemplifies this challenge. SAR vessel detection has previously been demonstrated only by ML models that either are too large for satellite deployment, have not been developed for sufficiently low-power hardware, or have only been tested on small SAR datasets that do not sufficiently represent the difficulty of the real-world task. Here we systematically explore a suite of architectural adaptations to develop a novel YOLOv8 architecture optimized for this task and FPGA-based processing. We deploy our model on a Kria KV260 MPSoC, and show it can analyze a ~700 megapixel SAR image in less than a minute, within common satellite power constraints (<10W). Our model has detection and classification performance only ~2% and 3% lower than values from state-of-the-art GPU-based models on the largest and most diverse open SAR vessel dataset, xView3-SAR, despite being ~50 and ~2500 times more computationally efficient. This work represents a key contribution towards on-satellite ML for time-critical SAR analysis, and more autonomous, scalable satellites.