We construct an infinite-dimensional compact metric space X, which is a closed subset of S x H, where S is the unit circle and H is the Hilbert cube, and a skew-product map F acting on X such that (X, F) is Li-Yorke sensitive but possesses at most countable scrambled sets. This disproves the conjecture of Akin and Kolyada that Li-Yorke sensitivity implies Li-Yorke chaos [Akin and Kolyada. Li-Yorke sensitivity. Nonlinearity 16, (2003), 1421-1433].