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

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.