  The following well known open problem is answered in the negative: Given two
compact spaces $X$ and $Y$ that admit minimal homeomorphisms, must the
Cartesian product $X\times Y$ admit a minimal homeomorphism as well? A key
element of our construction is an inverse limit approach inspired by
combination of a technique of Aarts & Oversteegen and the construction of
Slovak spaces by Downarowicz & Snoha & Tywoniuk. This approach allows us also
to prove the following result. Let $\phi\colon M\times\mathbb{R}\to M$ be a
continuous, aperiodic minimal flow on the compact, finite--dimensional metric
space $M$. Then there is a generic choice of parameters $c\in\mathbb{R}$, such
that the homeomorphism $h(x)=\phi(x,c)$ admits a noninvertible minimal map
$f\colon M\to M$ as an almost 1-1 extension.
