  We obtain a refinement of a selection principle for $(\mathcal{K},
\lambda)$-wide-$(s)$ sequences in Banach spaces due to Rosenthal. This result
is then used to show that if $C$ is a bounded, non-weakly compact, closed
convex subset of a Banach space $X$, then there exists a Hausdorff vector
topology $\tau$ on $X$ which is weaker than the weak topology, a closed, convex
$\tau$-compact subset $K$ of $C$ and an affine bi-Lipschitz map $T: K\to K$
without fixed points.
