Continual learning with vision-language models like CLIP offers a pathway toward scalable machine learning systems by leveraging its transferable representations. Existing CLIP-based methods adapt the pre-trained image encoder by adding multiple sets of learnable parameters, with each task using a partial set of parameters. This requires selecting the expected parameters for input images during inference, which is prone to error that degrades performance. To address this problem, we introduce LADA (Label-specific ADApter). Instead of partitioning parameters across tasks, LADA appends lightweight, label-specific memory units to the frozen CLIP image encoder, enabling discriminative feature generation by aggregating task-agnostic knowledge. To prevent catastrophic forgetting, LADA employs feature distillation for seen classes, preventing their features from being interfered with by new classes. Positioned after the image encoder, LADA prevents gradient flow to the frozen CLIP parameters, ensuring efficient training. Extensive results show that LADA achieves state-of-the-art performance in continual learning settings. The implementation code is available at https://github.com/MaolinLuo/LADA.