  Motivated by the Lipschitz-lifting property of Banach spaces introduced by
Godefroy and Kalton, we consider the lattice-lifting property, which is an
analogous notion within the category of Banach lattices and lattice
homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting
property if every lattice homomorphism to $X$ having a bounded linear
right-inverse must have a lattice homomorphism right-inverse. In terms of free
Banach lattices, this can be rephrased into the following question: which
Banach lattices embed into the free Banach lattice which they generate as a
lattice-complemented sublattice? We will provide necessary conditions for a
Banach lattice to have the lattice-lifting property, and show that this
property is shared by Banach spaces with a $1$-unconditional basis as well as
free Banach lattices. The case of $C(K)$ spaces will also be analyzed.
