  We study the projectivity of the free Banach lattice generated by a lattice
$\mathbb{L}$ in two cases: when the lattice is finite, and when the lattice is
an infinite linearly ordered set. We prove that in the first case it is
projective while in the second case, if the linear order contains either an
increasing sequence without upper bounds or a decreasing sequence without lower
bounds, then it is not projective.
