  We prove that every measurable function $f:\,[0,a]\to\mathbb{C}$ such that
$|f|=1$ a.e. on $[0,a]$ is an extreme point of the unit ball of the Lorentz
space $\Lambda(\varphi)$ on $[0,a]$ whenever $\varphi$ is a not linear,
strictly increasing, concave, continuous function on $[0,a]$ with
$\varphi(0)=0$. As a consequence, we complement the classical de Leeuw-Rudin
theorem on a description of extreme points of the unit ball of $H^1$ showing
that $H^1$ is a unique Hardy-Lorentz space $H(\Lambda(\varphi))$, for which
every extreme point of the unit ball is a normed outer function. Moreover,
assuming that $\varphi$ is strictly increasing and strictly concave, we prove
that every function $f\in H(\Lambda(\varphi))$,
$\|f\|_{H(\Lambda(\varphi))}=1$, such that the absolute value of its
nontangential limit ${f}(e^{it})$ is a constant on some set of positive measure
of $[0,2\pi]$, is an extreme point of the unit ball of $H(\Lambda(\varphi))$.
