While Large Language Models (LLMs) can serve as agents to simulate human behaviors (i.e., role-playing agents), we emphasize the importance of point-in-time role-playing. This situates characters at specific moments in the narrative progression for three main reasons: (i) enhancing users' narrative immersion, (ii) avoiding spoilers, and (iii) fostering engagement in fandom role-playing. To accurately represent characters at specific time points, agents must avoid character hallucination, where they display knowledge that contradicts their characters' identities and historical timelines. We introduce TimeChara, a new benchmark designed to evaluate point-in-time character hallucination in role-playing LLMs. Comprising 10,895 instances generated through an automated pipeline, this benchmark reveals significant hallucination issues in current state-of-the-art LLMs (e.g., GPT-4o). To counter this challenge, we propose Narrative-Experts, a method that decomposes the reasoning steps and utilizes narrative experts to reduce point-in-time character hallucinations effectively. Still, our findings with TimeChara highlight the ongoing challenges of point-in-time character hallucination, calling for further study.