Deep learning (DL) has emerged as a powerful tool for Synthetic Aperture Radar (SAR) ship classification. This survey comprehensively analyzes the diverse DL techniques employed in this domain. We identify critical trends and challenges, highlighting the importance of integrating handcrafted features, utilizing public datasets, data augmentation, fine-tuning, explainability techniques, and fostering interdisciplinary collaborations to improve DL model performance. This survey establishes a first-of-its-kind taxonomy for categorizing relevant research based on DL models, handcrafted feature use, SAR attribute utilization, and the impact of fine-tuning. We discuss the methodologies used in SAR ship classification tasks and the impact of different techniques. Finally, the survey explores potential avenues for future research, including addressing data scarcity, exploring novel DL architectures, incorporating interpretability techniques, and establishing standardized performance metrics. By addressing these challenges and leveraging advancements in DL, researchers can contribute to developing more accurate and efficient ship classification systems, ultimately enhancing maritime surveillance and related applications.