  We study the notion of permutation stability (or P-stability) for countable
groups. Our main result provides a wide class of non-amenable product groups
which are not P-stable. This class includes the product group
$\Sigma\times\Lambda$, whenever $\Sigma$ admits a non-abelian free quotient and
$\Lambda$ admits an infinite cyclic quotient. In particular, we obtain that the
groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are
not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that
P-stability is not closed under the direct product construction, which answers
a question of Becker, Lubotzky and Thom. The proof of our main result relies on
a construction of asymptotic homomorphisms from $\Sigma\times\Lambda$ to finite
symmetric groups starting from sequences of finite index subgroups in $\Sigma$
and $\Lambda$ with and without property $(\tau)$. Our method is sufficiently
robust to show that the groups covered are not even flexibly P-stable, thus
giving the first such non-amenable residually finite examples.
