Prompt learning for pre-trained Vision-Language Models (VLMs) like CLIP has demonstrated potent applicability across diverse downstream tasks. This lightweight approach has quickly gained traction from federated learning (FL) researchers who seek to efficiently adapt VLMs to heterogeneous scenarios. However, current federated prompt learning methods are habitually restricted to the traditional FL paradigm, where the participating clients are generally only allowed to download a single globally aggregated model from the server. While justifiable for training full-sized models under federated settings, in this work, we argue that this paradigm is ill-suited for lightweight prompts. By facilitating the clients to download multiple pre-aggregated prompts as fixed non-local experts, we propose Personalized Federated Mixture of Adaptive Prompts (pFedMoAP), a novel FL framework that personalizes the prompt learning process through the lens of Mixture of Experts (MoE). pFedMoAP implements a local attention-based gating network that learns to generate enhanced text features for better alignment with local image data on the client, benefiting from both local and downloaded non-local adaptive prompt experts. The non-local experts are sparsely selected from a server-maintained pool, fostering collaborative learning across clients. To evaluate the proposed algorithm, we conduct extensive experiments across 9 datasets under various heterogeneous federated settings. The results show that pFedMoAP consistently outperforms the state-of-the-art alternatives, underscoring its efficacy in personalizing prompt learning for CLIP within the federated learning paradigm.