Kottman proved that in the unit ball of any infinite-dimensional Banach space one can find an infinite subset such that the distance between any two distinct elements is larger than 1. Elton and Odell, employing Ramsey theory, strengthened this result by showing that it is possible to have all distances at least equal to 1+c for some c>0 depending only on the given space. Both these results are far-reaching generalizations of the Riesz lemma. We will investigate their non-separable versions focusing on two results: (1) the analogue of Kottman's theorem is valid for every non-separable reflexive Banach space (the set from the assertion is then uncountable); (2) the analogue of the Elton-Odell theorem is valid for every non-separable superreflexive Banach space.