Uncountable sets of unit vectors that are separated by more than 1

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal{A}\subset X$ of unit vectors such that $\mathrm{\parallel }x-y\mathrm{\parallel }>1$ for distinct $x,y\in \mathcal{A}$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\mathrm{\parallel }x-y\mathrm{\parallel }⩾1+\epsilon$ for some $\epsilon >0$ that depends only on $X$. If $K$ is a non-metrisable compact, Hausdorff space, then the unit sphere of $X=C(K)$ also contains such a subset; if moreover $K$ is perfectly normal, then one can find such a set with cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.