  We study the McKay correspondence for representations of the cyclic group of
order $p$ in characteristic $p$. The main tool is the motivic integration
generalized to quotient stacks associated to representations. Our version of
the change of variables formula leads to an explicit computation of the stringy
invariant of the quotient variety. A consequence is that a crepant resolution
of the quotient variety (if any) has topological Euler characteristic $p$ like
in the tame case. Also, we link a crepant resolution with a count of
Artin-Schreier extensions of the power series field with respect to weights
determined by ramification jumps and the representation.
