  We determine the minimal number of variables $\Gamma^*(d, K)$ which
guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$
odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We
prove that if $K$ is one of $\{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}),
\mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\}$, $\Gamma^*(d,K) =
\frac{3}{2}d$, and if $K$ is one of $\{\mathbb{Q}_2(\sqrt{-1}),
\mathbb{Q}_2(\sqrt{-5})\}$, $\Gamma^*(d,K) = d+1$. The case $d=6$ was
previously known.
