Naive vs. genuine A¹-connectedness

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.