  We show that recent results on adiabatic theory for interacting gapped
many-body systems on finite lattices remain valid in the thermodynamic limit.
More precisely, we prove a generalised super-adiabatic theorem for the
automorphism group describing the infinite volume dynamics on the quasi-local
algebra of observables. The key assumption is the existence of a sequence of
gapped finite volume Hamiltonians which generates the same infinite volume
dynamics in the thermodynamic limit. Our adiabatic theorem holds also for
certain perturbations of gapped ground states that close the spectral gap (so
it is an adiabatic theorem also for resonances and in this sense
`generalised'), and it provides an adiabatic approximation to all orders in the
adiabatic parameter (a property often called `super-adiabatic'). In addition to
existing results for finite lattices, we also perform a resummation of the
adiabatic expansion and allow for observables that are not strictly local.
Finally, as an application, we prove the validity of linear and higher order
response theory for our class of perturbations also for infinite systems.
  While we consider the result and its proof as new and interesting in itself,
they also lay the foundation for the proof of an adiabatic theorem for systems
with a gap only in the bulk, which will be presented in a follow-up article.
