Not all carcinogens are mutagens. Alcohol and estrogen, for example, do not damage DNA.
It's one of the assumptions of the Ames test that mutagenicity implies carcinogenicity, but is this always the case? I assumed that it was, but then I saw one of the comments here. I did some more research but the internet seems to be reluctant to be definitive on the subject. This guy claims no, but I'd prefer sources or at least a response that handles counterexamples like HPV. This paper claims yes, but doesn't list any specific examples. Some mutagens might be more specific to genes involved in cell cycle regulation, so I could see how a weak mutagen is a powerful carcinogen.
My question is, can you go the other way? Are there mutagens that just do not cause cancer? If they do not exist or are not known to exist, are they even possible?