We show that every orientable infinite-type surface without boundary is properly rigid as a consequence of a more general result.
Namely, we prove that if a homotopy equivalence $f\colon\ \Sigma'\to \Sigma$ between any two non-compact orientable surfaces without boundary is a proper map,
then $f$ is properly homotopic to a homeomorphism, provided $\Sigma$ is homeomorphic to neither the plane nor the punctured plane.
Thus, all non-compact orientable surfaces without boundary, except the plane and the punctured plane, are topologically rigid in a strong sense.