  Due to Narkiewicz a field $F$ has property (P) if for no polynomial $f\in
F[x]$ of degree at least two there is an infinite $f$-invariant subset of $F$.
We present a new example of an algebraic extension of $\mathbb{Q}$ satisfying
(P). This is the first example in which we can find points of arbitrarily small
positive Weil-height. Moreover, we study the possibility of property (P) for
the field generated by all symmetric Galois extensions of $\mathbb{Q}$. In
particular we prove that there are no infinite backward orbits of non linear
polynomials in this field.
