  Let $G$ be a transitive permutation group on a finite set of size at least
$2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a
derangement of prime power order. In this paper, we study the finite primitive
permutation groups with the extremal property that the order of every
derangement is an $r$-power, for some fixed prime $r$. First we show that these
groups are either almost simple or affine, and we determine all the almost
simple groups with this property. We also prove that an affine group $G$ has
this property if and only if every two-point stabilizer is an $r$-group. Here
the structure of $G$ has been extensively studied in work of Guralnick and
Wiegand on the multiplicative structure of Galois field extensions, and in
later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs.
