Starting from a comment on this ELL question, it appears to be a fairly common internet fact.
The most famous case involving beavers began when Francois de Laval, the first Bishop of Quebec, wrote to theologians at the Sorbonne (not the Vatican!), asking whether his flock could eat beaver during Lent. The theologians wrote back that yes, beavers could be considered fish.
Apparently, red meat is forbidden on Fridays in Lent. In order to circumvent this, a Canadian bishop François de Laval requested that beavers be considered fish and be eaten during Lent.
To me, this sounds like an urban myth to make the church (or Canadians) look silly. The pattern I've noticed is that websites discussing the beaver story mention de Laval, but sites about de Laval make no mention of the beaver story. Snopes forum cites Wikipedia, who cites the 1911 Encyclopedia Brittanica, but the article makes no mention of eating beaver other than it allegedly tastes like pork.
Did this ever happen? Is there a first-hand account my googling missed? Any Canadian Catholics have direct experience with this?