  It was shown by A. Beauville that if the canonical map $\varphi_{|K_M|}$ of a
complex smooth projective surface $M$ is generically finite, then ${\rm
deg}(\varphi_{|K_M|})\leq 36$. The first example of a surface with canonical
degree 36 was found by the second author. In this article, we show that for any
surface which is a degree four Galois \'etale cover of a fake projective plane
$X$ with the largest possible automorphism group ${\rm Aut}(X)=C_7:C_3$ (the
unique non-abelian group of order 21), the base locus of the canonical map is
finite, and we verify that 35 of these surfaces have maximal canonical degree
36. We also classify all smooth degree four Galois \'etale covers of fake
projective planes, which give possible candidates for surfaces of canonical
degree $36$. Finally, we also confirm in this paper the optimal upper bound of
the canonical degree of smooth threefolds of general type with sufficiently
large geometric genus, related to earlier work of C. Hacon and J.-X. Cai.
