Let $F = \mathbb{F}_p$. Then as $F$ is perfect every finite extension of $F$ is perfect. If $K$ is an extension of $F$ and $K$ is algebraically closed then of course $K$ is perfect.
Are there perfect extensions $K$ of $F$ with $trdeg_F K > 0$ other than algebraically closed fields?
I assume you mean that $F$ is the finite field with $p$ elements.
Take $K$ to be the field $F(T, T^{1/p}, T^{1/p^2}, \dots)$ obtained by formally adjoining a parameter $T$ and all of its $p$-power roots.