Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map f:M→S^n from a closed connected Riemannian spin manifold M with scalar curvature scal≥n(n−1) to the standard sphere Sn is an isometry if the degree of f is nonzero. We investigate if one can replace the condition deg(f)≠0 by the weaker condition that f is surjective. The answer turns out to be "no" for n≥3 but "yes" for n=2. If we replace the scalar curvature by Ricci curvature, the answer is "yes" in all dimensions.