Last year has witnessed the considerable interest of Large Language Models (LLMs) for their potential applications in recommender systems, which may mitigate the persistent issue of data sparsity. Though large efforts have been made for user-item graph augmentation with better graph-based recommendation performance, they may fail to deal with the dynamic graph recommendation task, which involves both structural and temporal graph dynamics with inherent complexity in processing time-evolving data. To bridge this gap, in this paper, we propose a novel framework, called DynLLM, to deal with the dynamic graph recommendation task with LLMs. Specifically, DynLLM harnesses the power of LLMs to generate multi-faceted user profiles based on the rich textual features of historical purchase records, including crowd segments, personal interests, preferred categories, and favored brands, which in turn supplement and enrich the underlying relationships between users and items. Along this line, to fuse the multi-faceted profiles with temporal graph embedding, we engage LLMs to derive corresponding profile embeddings, and further employ a distilled attention mechanism to refine the LLM-generated profile embeddings for alleviating noisy signals, while also assessing and adjusting the relevance of each distilled facet embedding for seamless integration with temporal graph embedding from continuous time dynamic graphs (CTDGs). Extensive experiments on two real e-commerce datasets have validated the superior improvements of DynLLM over a wide range of state-of-the-art baseline methods.