Solubility of additive sextic forms over ramified quadratic extensions of $Q_2$

In this article, we study the equation $a_1x_1^6+a_2x_2^6+\cdots+a_sx_s^6=0$ over the six ramified quadratic extensions of the $p$-adic field $\mathbb{Q}_2$. For all of these extensions, we show that if $s\geq 9$, then this equation has a nontrivial solution regardless of the values of the coefficients. For four of the extensions, we show that $9$ is the smallest number of variables that guarantees that the equation will have a nontrivial solution.