  We study the fixed points of the Berezin transform in polyanalytic Fock
spaces of $\mathbb{C}$. We show that an $L^p$ function, $p\in[1,+\infty]$, with
respect to the Lebesgue measure is invariant under this transformation if and
only if it is harmonic. From this we deduce that the only bounded fixed points
of the Berezin transform of polyanalytic Fock spaces are constant functions.
