  We construct a nonseparable Banach space $\mathcal X$ (actually, of density
continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of
$\mathcal X$ contains uncountably many points distant by less than $1$ (in
fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in
the negative the central problem of the search for a nonseparable version of
Kottman's theorem which so far has produced many deep positive results for
special classes of Banach spaces and has related the global properties of the
spaces to the distances between points of uncountable subsets of the unit
sphere.
  The property of our space is strong enough to imply that it contains neither
an uncountable Auerbach system nor an uncountable equilateral set. The space is
a strictly convex renorming of the Johnson-Lindenstrauss space induced by an
$\mathbb R$-embeddable almost disjoint family of subsets of $\mathbb N$. We
also show that this special feature of the almost disjoint family is essential
to obtain the above properties.
