  First we prove that if a separable Banach space $X$ contains an isometric
copy of an infinite-dimensional space $A(S)$ of affine continuous functions on
a Choquet simplex $S$, then its dual $X^*$ lacks the weak$^*$ fixed point
property for nonexpansive mappings. Then, we show that the dual of a separable
Lindenstrauss space $X$ fails the weak$^*$ fixed point property for
nonexpansive mappings if and only if $X$ has a quotient isometric to some space
$A(S)$. Moreover, we provide an example showing that "quotient" cannot be
replaced by "subspace". Finally, it is worth to be mentioned that in our
characterization the space $A(S)$ cannot be substituted by any space
$\mathcal{C}(K)$ of continuous functions on a compact Hausdorff $K$.
