  We generalize a classical extension result by Seeley in the context of
Bastiani's differential calculus to infinite dimensions. The construction
follows Seeley's original approach, but is significantly more involved as not
only $C^k$-maps (for $k\in \mathbb{N}\cup\{\infty\}$) on (subsets of) half
spaces are extended, but also continuous extensions of their differentials to
some given piece of boundary of the domains under consideration. A further
feature of the generalization is that we construct families of extension
operators (instead of only one single extension operator) that fulfill certain
compatibility (and continuity) conditions. Various applications are discussed
as well.
