  We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of
finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness
preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by
constraining the componentwise behavior of the inverse $g=f^{-1}$ with a
theorem admitting a graph-theoretic interpretation. We also show that if $X, Y$
are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X \to
B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type
$B_{X'} \to B_{X'}$ for some Banach space $X'$, and that a homeomorphic
1-Lipschitz non-isometry of type $B_X \to B_X$ restricts to a homeomorphic
1-Lipschitz non-isometry of type $B_S \to B_S$ for some separable subspace $S
\leq X$.
