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

A compact minimal space $Y$ such that its square $Y\times Y$ is not minimal