Summary: In this article, we study statistical attractors of skew products which have an

$m$-dimensional compact manifold

$M$ as a fiber and their

$\u03f5$-invisible subsets. For any

$n\ge 100{m}^{2},m=dim\left(M\right)$, we construct a set

${\mathcal{R}}_{n}$ in the space of skew products over the horseshoe with the fiber

$M$ having the following properties. Each

${C}^{2}$-skew product from possesses a statistical attractor with an

$\u03f5$-invisible part, for an extraordinary value of

$\u03f5(\u03f5={(m+1)}^{-n})$, whose size of invisibility is comparable to that of the whole attractor, and the Lipschitz constants of the map and its inverse are no longer than

$L$. The set

${\mathcal{R}}_{n}$ is a ball of radius

$O\left({n}^{-3}\right)$ in the space of skew products over the horseshoe with the

${C}^{1}$-metric. In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Moreover, for skew products which have an m-sphere as a fiber, it consists of structurally stable skew products. Our construction develops the example of

*Yu. Ilyashenko* and

*A. Negut* [Nonlinearity 23, No. 5, 1199–1219 (2010;

Zbl 1204.37017)] to skew products which have an m-dimensional compact manifold as a fiber,

$m\ge 2$.