  In 2022, Hatori gave a sufficient condition for complex Banach spaces to have
the complex Mazur--Ulam property. In this paper, we introduce a class of
complex Banach spaces $B$ that do not satisfy the condition but enjoy the
property that every surjective isometry on the unit sphere of such $B$ admits
an extension to a surjective real linear isometry on the whole space $B$.
Typical examples of Banach spaces studied in this note are the spaces ${\rm
Lip}([0,1])$ of all Lipschitz complex-valued functions on $[0,1]$ and
$C^1([0,1])$ of all continuously differentiable complex-valued functions on
$[0,1]$ equipped with the norm $|f(0)|+\|f'\|_\infty$.
