The answer is yes, and the issue is discussed in detail in $\S 3.5-3.6$ of these notes from a recent graduate number theory course.
It is very much as Akhil suggests: the key idea is Krasner's Lemma (introduced and explained in my notes). However, as Krasner's Lemma pertains to separable extensions, there is a little further work that needs to be done in positive characteristic: how do we know that $\overline{F}$ is algebraically closed rather than just separably closed?
The answer is given by the following fact (Proposition 27 on p. 15 of my notes):
A field which is separably closed and complete with respect to a nontrivial valuation is algebraically closed.
The idea of the proof is to approximate a purely inseparable extension by a sequence of (necessarily separable) Artin-Schreier extensions. I should probably also mention that I found this argument in some lecture notes of Brian Conrad (and I haven't yet found it in the standard texts on the subject).
Note that Corollary 28 (i.e., the very next result) is the answer to your question.