The quality and realism of synthetically generated fingerprint images have increased significantly over the past decade fueled by advancements in generative artificial intelligence (GenAI). This has exacerbated the vulnerability of fingerprint recognition systems to data injection attacks, where synthetic fingerprints are maliciously inserted during enrollment or authentication. Hence, there is an urgent need for methods to detect if a fingerprint image is real or synthetic. While it is straightforward to train deep neural network (DNN) models to classify images as real or synthetic, often such DNN models overfit the training data and fail to generalize well when applied to synthetic fingerprints generated using unseen GenAI models. In this work, we formulate synthetic fingerprint detection as a continual few-shot adaptation problem, where the objective is to rapidly evolve a base detector to identify new types of synthetic data. To enable continual few-shot adaptation, we employ a combination of binary cross-entropy and supervised contrastive (applied to the feature representation) losses and replay a few samples from previously known styles during fine-tuning to mitigate catastrophic forgetting. Experiments based on several DNN backbones (as feature extractors) and a variety of real and synthetic fingerprint datasets indicate that the proposed approach achieves a good trade-off between fast adaptation for detecting unseen synthetic styles and forgetting of known styles.