  In this paper we study the $w^*$-fixed point property for nonexpansive
mappings. First we show that the dual space $X^*$ lacks the $w^*$-fixed point
property whenever $X$ contains an isometric copy of the space $c$. Then, the
main result of our paper provides several characterizations of weak-star
topologies that fail the fixed point property for nonexpansive mappings in
$\ell_1$ space. This result allows us to obtain a characterization of all
separable Lindenstrauss spaces $X$ inducing the failure of $w^*$-fixed point
property in $X^*$.
