As an addendum to Jochen Wengenroths answer: If you are willing to restrict your choice of compact sets somewhat (to those whose interior is dense), then you might find the answers to your questions (for E finite dimensional) in the recent preprint https://arxiv.org/pdf/2006.00254.pdf Note that the restriction to finite dimensionsl spaces here is necessary as on one hand there are no compact sets with nonempty interior in infinite dimensional spaces and in addition, the differentiability discussed in the paper contains with the Frechet differentiability you asked for on finite dimensional spaces.

As an addendum to Jochen Wengenroth's answer: If you are willing to restrict your choice of compact sets somewhat (to those whose interior is dense), then you might find the answers to your questions (for $E$ finite dimensional) in the recent preprint

• Helge Glockner, Smoothing operators for vector-valued functions and extension operators, arXiv:2006.00254.

Note that the restriction to finite-dimensional spaces here is necessary as on one hand there are no compact sets with nonempty interior in infinite-dimensional spaces and in addition, the differentiability discussed in the paper contains with the Fréchet differentiability you asked for on finite-dimensional spaces.