Yes but they are a different species. North America has the appropriately named North American beaver whereas they have the Eurasian beaver.
Eurasian beavers were actually nearly hunted to extinction by the onset of the 20th century but fortunately conservation efforts have led to their population numbers rebounding considerably.
Wait, then why did Europeans have such a raging boner for beaver pelts from the new world during the colonial period? Couldn't they just.... Get them from these local beavers?
Germany has taken some serious "renaturation" steps to fix the grave errors done since the 1940s to 1980s where they straightened rivers and dried up a lot of land. E.g. at all of Altmühl's weirs they dug a little creek around it to enable fish to migrate or sometimes even bigger projects like this one: