Let u be any element in F and restrict attention to K(u). In fact let F = K(u). If this is separable then we are done.
The extension K(u) includes all the conjugates of u, and these conjugates are distinct. The common multiplicity is 1. this is proved elsewhere. therefore u is separable, and every finite field is perfect.
If you're looking for an inseparable extension, you need to start with the polynomials over Zp. this is a transcendental extension of Zp, and when it acts as a base, it just might lead to some inseparable extensions.