Square-free values of polynomials evaluated at primes over a function field

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.