How far is an extension of $p$-adic fields from having a normal integral basis?

Let $L\mathrm{/}K$ be a finite Galois extension of $p$-adic fields with group $G$. It is well-known that ${\mathcal{O}}_L$ contains a free ${\mathcal{O}}_K[G]$-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree $p$ of $p$-adic fields.