  In this paper, we study the property of weak approximation with Brauer-Manin
obstruction for surfaces with respect to field extensions of number fields. For
any nontrivial extension of number fields L/K, assuming a conjecture of M.
Stoll, we construct a smooth, projective, and geometrically connected surface
over K such that it satisfies weak approximation with Brauer-Manin obstruction
off all archimedean places, while its base change to L fails. Then we
illustrate this construction with an explicit unconditional example.
