Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special maximal subgroups of such a group are.

More specifically, let $\mathbf{K}$ be the subgroup consisting of unitary matrices whose entries lie in the ring of integers $\mathfrak{o}_{L'}$. If the residue characteristic of $L'/L$ is not $2$, and $L'/L$ is unramified, then $\mathbf{K}$ is hyperspecial. However, when the residue characteristic is $2$, then $\mathbf{K}$ is not even maximal when $n = 3$.

My question is:

When $L'/L$ is ramified and has residue characteristic not $2$, is $\mathbf{K}$ still maximal special (even though it is not hyperspecial)? If not, what are some of the special subgroups?

If the residue characteristic is $2$, when is $\mathbf{K}$ special? When it isn't, can we find a special subgroup containing it?