On projective Banach lattices of the form C(K) and FBL[E]

We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of $c_0$ must contain an ${\mathrm{\ell }}_1$-subsequence. As a consequence, if Banach lattices ${\mathrm{\ell }}_p$ or $FBL[E]$ are projective, then $p=1$ or $E$ has the Schur property, respectively. On the other hand, we show that $C(K)$ is projective whenever $K$ is an absolute neighbourhood retract, answering a question by de Pagter and Wickstead.