  We introduce a new class of Banach spaces, called generalized-lush spaces
(GL-spaces for short), which contains almost-CL-spaces, separable lush spaces
(specially, separable $C$-rich subspaces of $C(K)$), and even the
two-dimensional space with hexagonal norm. We obtain that the space $C(K,E)$ of
the vector-valued continuous functions is a GL-space whenever $E$ is, and show
that the GL-space is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an
application, we prove that the Mazur-Ulam property holds for a larger class of
Banach spaces, called local-GL-spaces, including all lush spaces and GL-spaces.
Furthermore, we generalize the stability properties of GL-spaces to
local-GL-spaces. From this, we can obtain many examples of Banach spaces having
the Mazur-Ulam property.
