Personal Narratives (PN) - recollections of facts, events, and thoughts from one's own experience - are often used in everyday conversations. So far, PNs have mainly been explored for tasks such as valence prediction or emotion classification (i.e. happy, sad). However, these tasks might overlook more fine-grained information that could nevertheless prove relevant for understanding PNs. In this work, we propose a novel task for Narrative Understanding: Emotion Carrier Recognition (ECR). We argue that automatic recognition of emotion carriers, the text fragments that carry the emotions of the narrator (i.e. 'loss of a grandpa', 'high school reunion'), from PNs, provides a deeper level of emotion analysis needed, for instance, in the mental healthcare domain. In this work, we explore the task of ECR using a corpus of PNs manually annotated with emotion carriers and investigate different baseline models for the task. Furthermore, we propose several evaluation strategies for the task. Based on the inter-annotator agreement, the task in itself was found to be complex and subjective for humans. Nevertheless, we discuss evaluation metrics that could be suitable for applications based on ECR.