Let G be a Lie group, Γ ⊂ G a discrete subgroup, X = G/Γ, and f an affine map from X to itself. We give conditions on a submanifold Z of X guaranteeing that the set of points x ∈ X with f -trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.