  We compute the precise value of the measure of noncompactness of Sobolev
embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on
strip-like domains $D$ of the form
$\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. We show that such
embeddings are always maximally noncompact, that is, their measure of
noncompactness coincides with their norms. Furthermore, we show that not only
the measure of noncompactness but also all strict $s$-numbers of the embeddings
in question coincide with their norms. We also prove that the maximal
noncompactness of Sobolev embeddings on strip-like domains remains valid even
when Sobolev-type spaces built upon general rearrangement-invariant spaces are
considered. As a by-product we obtain the explicit form for the first
eigenfunction of the pseudo-$p$-Laplacian on an $n$-dimensional rectangle.
