No product of two non-trivial countable-dimensional continua maps lightly into any of the factors

We shall prove that if X, Y are compact metrizable spaces of positive dimension and h: X x Y --> X is a continuous map with zero-dimensional fibers then X contains a non-trivial continuum without one-dimensional subsets; in particular X is not a countable union of zero-dimensional sets, which provides a negative answer to a question of J. Dudák and B. Vejnar.