In people, the ability to solve analogies such as "body : feet :: table : ?" emerges in childhood, and appears to transfer easily to other domains, such as the visual domain "( : ) :: < : ?". Recent research shows that large language models (LLMs) can solve various forms of analogies. However, can LLMs generalize analogy solving to new domains like people can? To investigate this, we had children, adults, and LLMs solve a series of letter-string analogies (e.g., a b : a c :: j k : ?) in the Latin alphabet, in a near transfer domain (Greek alphabet), and a far transfer domain (list of symbols). Children and adults easily generalized their knowledge to unfamiliar domains, whereas LLMs did not. This key difference between human and AI performance is evidence that these LLMs still struggle with robust human-like analogical transfer.