I need an example of a p-adic field extention $L/F$ of degree $[L:F]=2n$ without a subfield $K\subset L$ of degree $[K:F] = 2$.

For every prime $p$, every local field $F$ of residual characteristic $p$, and every integer $n>0$, there is a separable extension $L$ of $F$ of degree $[L:F]=p^n$ which does not have any intermediate extensions. All these $L$ can be explicitly parametrised. See for example Wildly Primitive Extensions.

didanswer, I am reluctant to do that. $\endgroup$ – LSpice Aug 21 '19 at 16:33explicitpolynomial with Galois group $S_4$ over $\mathbf{Q}_2$, so $K = \mathbf{Q}_2[x]/(x^4+2x+2)$ is a degree $4$ extension with no quadratic subfields. $\endgroup$ – Puddles the turtle Aug 21 '19 at 23:46