Remark

Let $s$ be an integer $> 0$; the subfield $\mathbf{F}_q((T^s)) = L$ of $K = \mathbf{F}_q((T))$ is closed in $K$ and $e(K/L) = s$ and $f(K/L) = 1$. It is therefore seen that there are closed subfields $L$ of $K$ which are not discrete and such that $e(K/L)$ (and a fortiori the degree $[K:L]$) is arbitrarily large (contrary to what happens for locally compact fields of characteristic 0, where every locally compact subfield $L$ of such a field $K$ necessarily contains $\mathbf{R}$ or $\mathbf{Q}_p$ and therefore $[K:L]$ is bounded).