This article introduces a novel physical-layer (PHY-layer) authentication scheme, called transfer learning-based PHY-layer authentication (TL-PHA), aiming to achieve fast online user authentication that is highly desired for latency-sensitive applications such as edge computing. The proposed TL-PHA scheme is characterized by incorporating with a novel convolutional neural network architecture, namely, the triple-pool network (TP-Net), for achieving lightweight and online classification, as well as effective data augmentation methods for generation of data set samples for the network model training. To assess the performance of the proposed scheme, we conducted two sets of experiments, including the one using computer-simulated channel data and the other utilizing real experiment data generated by our wireless testbed. The results demonstrate the superiority of the proposed scheme in terms of authentication accuracy, detection rate, and training complexity compared to all the considered counterparts.