Social media contains unfiltered and unique information, which is potentially of great value, but, in the case of misinformation, can also do great harm. With regards to biomedical topics, false information can be particularly dangerous. Methods of automatic fact-checking and fake news detection address this problem, but have not been applied to the biomedical domain in social media yet. We aim to fill this research gap and annotate a corpus of 1200 tweets for implicit and explicit biomedical claims (the latter also with span annotations for the claim phrase). With this corpus, which we sample to be related to COVID-19, measles, cystic fibrosis, and depression, we develop baseline models which detect tweets that contain a claim automatically. Our analyses reveal that biomedical tweets are densely populated with claims (45 % in a corpus sampled to contain 1200 tweets focused on the domains mentioned above). Baseline classification experiments with embedding-based classifiers and BERT-based transfer learning demonstrate that the detection is challenging, however, shows acceptable performance for the identification of explicit expressions of claims. Implicit claim tweets are more challenging to detect.