  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
$\ell_1$-subsequence. As a consequence, if Banach lattices $\ell_p$ 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.
