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 $Xtimes Y$ 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 $phicolon Mtimesmathbb{R}to M$ be a continuous, aperiodic minimal flow on the compact, finite--dimensional metric space $M$. Then there is a generic choice of parameters $cinmathbb{R}$, such that the homeomorphism $h(x)=phi(x,c)$ admits a noninvertible minimal map $fcolon Mto M$ as an almost 1-1 extension.
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