It is well-known that, in a real-closed field $K$, every polynomial of degree $>2$ is reducible in $K$. But in this case the characteristic of $K$ is zero.

My question is: there exists a non-perfect field $F$ and a positive integer $n=n(F)$ such that all polynomials of degree $>n$ are reducible in $F$.

Edit: polynomials of degree $>n$ coprime with the characteristic $p>0$ of $F$