  We completely solve the problem whether the product of two compact metric
spaces admitting minimal maps also admits a minimal map. Recently Boro\'nski,
Clark and Oprocha gave a negative answer in the particular case when
homeomorphisms rather than continuous maps are considered. In the present paper
we show that there is a metric continuum $X$ admitting a minimal map, in fact a
minimal homeomorphism, such that $X\times X$ does not admit any minimal map.
