  In this article we deal with the free Banach lattice $FBL\langle \mathbb{L}
\rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle
\mathbb{L} \rangle$ is projective then $\mathbb{L}$ has a maximum and a
minimum. On the other hand, we show that if $\mathbb{L}$ has maximum and
minimum then $FBL\langle \mathbb{L} \rangle$ is $2$-lattice isomorphic to a
$C(K)$-space. As a consequence, $FBL\langle \mathbb{L} \rangle$ is projective
if and only if it is lattice isomorphic to a $C(K)$-space with $K$ being an
absolute neighborhood retract. As an application, we characterize those
linearly ordered sets and Boolean algebras for which the corresponding free
Banach lattice is projective.
