Deep learning (DL) has emerged as a powerful tool for synthetic aperture radar (SAR) ship classification. This survey comprehensively analyzes the various DL techniques used in this field. It highlights the importance of incorporating handcrafted features, leveraging public datasets, data augmentation, fine-tuning, explainability techniques, and interdisciplinary collaboration to improve DL model performance, and identifies key trends and challenges. This survey presents a first-of-its-kind taxonomy to categorize relevant research based on the impact of DL models, handcrafted feature usage, SAR attribute utilization, and fine-tuning. It discusses the methodologies and impact of various techniques used in SAR ship classification tasks. Finally, this survey explores potential directions for future research, including addressing data scarcity, exploring novel DL architectures, integrating interpretability techniques, and establishing standardized performance metrics. By addressing these challenges and leveraging advancements in DL, researchers can contribute to the development of more accurate and efficient ship classification systems, ultimately enhancing maritime surveillance and related applications.