  We study a function field version of a classical problem concerning
square-free values of polynomials evaluated at primes. We show that for a
square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density
as $n\to \infty$ of primes $P \in \mathbb{F}_q[t]$ of degree $n$ such that
$f(P)$ is square-free.
  Over the integers the analogous result is only known when all irreducible
factors of $f$ have degree at most 3.
