  We show that the Stokes operator defined on $\mathrm{L}^p_{\sigma} (\Omega)$
for an exterior Lipschitz domain $\Omega \subset \mathbb{R}^n$ $(n \geq 3)$
admits maximal regularity provided that $p$ satisfies $| 1/p - 1/2| < 1/(2n) +
\varepsilon$ for some $\varepsilon > 0$. In particular, we prove that the
negative of the Stokes operator generates a bounded analytic semigroup on
$\mathrm{L}^p_\sigma (\Omega)$ for such $p$. In addition,
$\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the Stokes semigroup and
its gradient with optimal decay estimates are obtained. This enables us to
prove the existence of mild solutions to the Navier--Stokes equations in the
critical space $\mathrm{L}^{\infty} (0 , T ; \mathrm{L}^3_{\sigma} (\Omega))$
(locally in time and globally in time for small initial data).
