Which case did Perry Mason lose?

Yes, he lost two cases. One was "The Case of the Deadly Verdict." 10/17/1963. But it wasn't his fault. His client withheld evidence needed to win.
He also lost "The Case of the Terrified Typist" (6/21/1958). After his client has been found guilty and sentenced to death, Mason discovers and reveals that his client isn't the person whose name (Jefferson) he was using throughout the trial, and successfully asks for a retrial since much of the evidence applied to the real Jefferson, not his client (real name Kincaid). Still, it would be wrong to say he "won in the end," since he admits that Kincaid is indeed guilty, and makes it clear that Burger should have no problem getting another guilty verdict against Kincaid (using the correct evidence and name).