  In this paper we classify complex K3 surfaces with non-symplectic
automorphism of order 8 that leaves invariant a smooth elliptic curve. We show
that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is
the disjoint union of elliptic curves, rational curves and points, whose number
does not exceed 1, 2, respectively 14. We give examples corresponding to
several types of fixed locus in the classification.
