We tackle continual adaptation of vision-language models to new attributes, objects, and their compositions in Compositional Zero-Shot Learning (CZSL), while preventing forgetting of prior knowledge. Unlike classical continual learning where classes are disjoint, CCZSL is more complex as attributes and objects may reoccur across sessions while compositions remain unique. Built on a frozen VLM backbone, we propose the first Prompt-based Continual Compositional Zero-Shot Learning (PromptCCZSL) framework that retains prior knowledge through recency-weighted multi-teacher distillation. It employs session-aware compositional prompts to fuse multimodal features for new compositions, while attribute and object prompts are learned through session-agnostic fusion to maintain global semantic consistency, which is further stabilized by a Cosine Anchor Loss (CAL) to preserve prior knowledge. To enhance adaptation in the current session, an Orthogonal Projection Loss (OPL) ensures that new attribute and object embeddings remain distinct from previous ones, preventing overlap, while an Intra-Session Diversity Loss (IDL) promotes variation among current-session embeddings for richer, more discriminative representations. We also introduce a comprehensive protocol that jointly measures catastrophic forgetting and compositional generalization. Extensive experiments on UT-Zappos and C-GQA benchmarks demonstrate that PromptCCZSL achieves substantial improvements over prior VLM-based and non-VLM baselines, setting a new benchmark for CCZSL in closed-world settings.