  We study the non-localization of extended worldsheet supersymmetry under
T-duality, when the associated complex structure depends on the coordinate with
respect to which duality is performed. First, the canonical transformation
which implements T-duality is generalized to the supersymmetric non-linear
$\sigma$-models. Then, we obtain the non-local object which replaces the
complex structure in the dual theory and write down the condition it should
satisfy so that the dual action is invariant under the non-local supersymmetry.
For the target space, this implies that the supersymmetry transformation
parameter is a non-local spinor. The analogue of the Killing equation for this
non-local spinor is obtained. It is argued that in the target space, the
supersymmetry is no longer realized in the standard way. The string theoretic
origin of this phenomenon is briefly discussed.
