  Background: For its simplicity, the eikonal method is the tool of choice to
analyze nuclear reactions at high energies ($E>100$ MeV/nucleon), including
knockout reactions. However, so far, the effective interactions used in this
method are assumed to be fully local.
  Purpose: Given the recent studies on non-local optical potentials, in this
work we assess whether non-locality in the optical potentials is expected to
impact reactions at high energies and then explore different avenues for
extending the eikonal method to include non-local interactions.
  Method: We compare angular distributions obtained for non-local interactions
(using the exact R-matrix approach for elastic scattering and the adiabatic
distorted wave approximation for transfer) with those obtained using their
local-equivalent interactions.
  Results: Our results show that transfer observables are significantly
impacted by non-locality in the high-energy regime. Because knockout reactions
are dominated by stripping (transfer to inelastic channels), non-locality is
expected to have a large effect on knockout observables too. Three approaches
are explored for extending the eikonal method to non-local interactions,
including an iterative method and a perturbation theory.
  Conclusions: None of the derived extensions of the eikonal model provide a
good description of elastic scattering. This work suggests that non-locality
removes the formal simplicity associated with the eikonal model.
