Every day, many people use social media platforms to share information, thoughts, narratives and personal experiences. The vast volume of user-generated content offers valuable insights into the latest news and trends but also poses serious challenges due to the presence of a lot of false information. In this paper we focus on analyzing the online conversation on Twitter to identify and unveil false information related to COVID-19. To address this challenge, we devised a semi-supervised approach that combines false information detection with a neural topic modeling algorithm. By leveraging a small amount of labeled data, a BERT-based classifier is fine-tuned on the false information detection task and then is used to annotate a large amount of COVID-related tweets, organized in a topic-based clustering structure. This approach allows for effectively identifying the degree of false information in each discussion topic related to COVID-19. Specifically, our approach allows for investigating the presence of false information from a topical perspective, enabling us to examine its impact on specific topics underlying the online discussion. Among the topics with the highest incidence of false information, we found allergic reactions, microchips in vaccines, and 5G- and lockdown-related conspiracy theories. Our findings highlight the importance of leveraging social media platforms as valuable sources of information but at the same time how essential it is to identify and mitigate the impact of false information in online communities.