  We prove that any Fourier--Mukai partner of an abelian surface over an
algebraically closed field of positive characteristic is isomorphic to a moduli
space of Gieseker-stable sheaves. We apply this fact to show that the
Fourier--Mukai set of canonical covers of hyperelliptic and Enriques surfaces
over an algebraically closed field of characteristic greater than three is
trivial. These results extend to positive characteristic earlier results of
Bridgeland--Maciocia and Sosna.
