Let $(X,d_X)$ be a pointed metric space with a basepoint designated by $e_X$, and let $\mathbb{K}$ be the field of real or complex numbers. The pointed Lipschitz space $\mathrm{Lip}_0(X)$ is the Banach space of all Lipschitz functions $f\colon X\to\mathbb{K}$ for which $f(e_X)=0$, endowed with the norm defined by $$ L(f)=\sup\left\{\frac{\left|f(x)-f(y)\right|}{d_X(x,y)}\colon x,y\in X,\;x\neq y\right\}. $$ We study 2-local isometries between spaces $\mathrm{Lip}_0(X)$. Namely, under the conditions of completeness and uniform concavity on the metric spaces $X$ and $Y$ - which are necessary to have a convenient description of the surjective linear isometries from $\mathrm{Lip}_0(X)$ to $\mathrm{Lip}_0(Y)$ - we obtain that every 2-local isometry $\Delta$ from $\mathrm{Lip}_0(X)$ to $\mathrm{Lip}_0(Y)$ admits a representation as the sum of a weighted composition operator and a homogeneous Lipschitz functional on, at least, a subspace $Y_0$ of $Y$ which is isometric to $Y$. Moreover, when $X$ is also separable, we prove that $Y_0$ coincides with $Y$, and thus $\Delta$ is both linear and surjective.