  We show that the triviality of sections of the sheaf of A^1-chain connected
components of a space over finitely generated separable field extensions of the
base field is not sufficient to ensure the triviality of the sheaf of its
A^1-chain connected components, contrary to the situation with genuine
A^1-connected components. As a consequence, we show that there exists an
A^1-connected scheme for which the Morel-Voevodsky singular construction is not
A^1-local.
