Handling unanswerable questions (UAQ) is crucial for LLMs, as it helps prevent misleading responses in complex situations. While previous studies have built several datasets to assess LLMs’ performance on UAQ, these datasets lack factual knowledge support, which limits the evaluation of LLMs’ ability to utilize their factual knowledge when handling UAQ. To address the limitation, we introduce a new unanswerable question dataset UAQFact, a bilingual dataset with auxiliary factual knowledge created from a Knowledge Graph. Based on UAQFact, we further define two new tasks to measure LLMs’ ability to utilize internal and external factual knowledge, respectively. Our experimental results across multiple LLM series show that UAQFact presents significant challenges, as LLMs do not consistently perform well even when they have factual knowledge stored. Additionally, we find that incorporating external knowledge may enhance performance, but LLMs still cannot make full use of the knowledge which may result in incorrect responses. Our code and dataset are available at https://github.com/cytan17726/UAQ_Fact.