The norm $\lVert\mathord{\cdot}\rVert^{\vee}$ is ultrametric, and $\lVert\mathord{\cdot}\rVert^{\vee\vee}=\lVert\mathord{\cdot}\rVert$ if and only if $\lVert\mathord{\cdot}\rVert$ is ultrametric. ([CMor18, Section 2.2.3])