Let
$(X,\,d)$ be a metric space, and let
$\text{Lip(}X\text{)}$ denote the Banach space of all scalar-valued bounded Lipschitz functions
$f$ on
$X$ endowed with one of the natural norms
$$\left\| f \right\|\,=\,\max \{{{\left\| f \right\|}_{\infty }},\,L(f)\}\,\,\text{or}\,\,\left\| f \right\|\,=\,{{\left\| f \right\|}_{\infty }}\,+\,L(f),$$
where
$L(f)$ is the Lipschitz constant of
$f$. It is said that the isometry group of
$\text{Lip(}X\text{)}$ is canonical if every surjective linear isometry of
$\text{Lip(}X\text{)}$ is induced by a surjective isometry of
$X$. In this paper we prove that if
$X$ is bounded separable and the isometry group of
$\text{Lip(}X\text{)}$ is canonical, then every 2-local isometry of
$\text{Lip(}X\text{)}$ is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of
$\text{Lip(}X\text{)}$ when
$X$ is bounded.