  We develop a functional framework suitable for the treatment of partial
differential equations and variational problems on evolving families of Banach
spaces. We propose a definition for the weak time derivative that does not rely
on the availability of a Hilbertian structure and explore conditions under
which spaces of weakly differentiable functions (with values in an evolving
Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions
compactness result is proved. We analyse concrete examples of function spaces
over time-evolving spatial domains and hypersurfaces for which we explicitly
provide the definition of the time derivative and verify isomorphism properties
with the aforementioned Sobolev--Bochner spaces. We conclude with the proof of
well posedness for a class of nonlinear monotone problems on an abstract
evolving space (generalising the evolutionary $p$-Laplace equation on a moving
domain or surface) and identify some additional problems that can be formulated
with the setting developed in this work.
