Can a bleeding ulcer lead to anemia over time?

Yes. Anemia is a low blood count. If you continue to bleed from an ulcer into the gut, this causes blood loss which lowers the blood count causing anemia.

Agree with my colleague, but wish to add that a bigger risk could eventuate from an upper GI bleed, which could be associated with ulcer perforation or penetration, and this would develop if timely intervention were not undertaken. If there is suspicion of peptic ulcer disease, address ASAP with your doctor.