  We construct extended TQFTs associated to Rozansky--Witten models with target
manifolds $T^*\mathbb{C}^n$. The starting point of the construction is the
3-category whose objects are such Rozansky--Witten models, and whose morphisms
are defects of all codimensions. By truncation, we obtain a (non-semisimple)
2-category $\mathcal{C}$ of bulk theories, surface defects, and isomorphism
classes of line defects. Through a systematic application of the cobordism
hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in
$\mathcal{C}$ for every affine Rozansky--Witten model. By evaluating this TQFT
on closed surfaces we obtain the infinite-dimensional state spaces (graded by
flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we
explicitly compute the commutative Frobenius algebras that classify the
restrictions of the extended theories to circles and bordisms between them.
