Let \(k\) be a dyadic local field of characteristic \(0\). Using the reduced norm of elements in \(D\), one can extend the discrete valuation on \(k\) to a surjective valuation \(\nu:D\to\mathbb{Z}\cup\infty\) and obtains thus the unique maximal order \(\mathcal{O}_D=\{ x\in D\,|\,\nu(x)\geq 0\}\). In this paper, the authors study full \(\mathcal{O}_D\)-lattices \(\Lambda\) in a finite dimensional skew hermitian space \((V,h)\) over \(D\) with respect to the standard involution on \(D\). Let \(\mathcal{U}_k^+(\Lambda)\) denote the stabilizer of \(\Lambda\) in the special unitary group \(\mathcal{U}_k^+\)of \(h\), and let \(\theta: \mathcal{U}_k^+\to k^*/k^{*2}\) denote the spinor norm. The spinor image \(H(\Lambda)\subseteq k^*\) is then defined by the relation \(H(\Lambda)/k^{*2}=\theta(\mathcal{U}_k^+(\Lambda))\). Completing earlier results by the first author [Contemp. Math. 344, 19–29 (2004; Zbl 1152.11325); Arch. Math. 94, No. 4, 351–356 (2010; Zbl 1193.11034)], the present paper provides an explicit computation of \(H(\Lambda)\) in the case \(k=\mathbb{Q}_2\). The results depend on the orthogonal decomposition of \(\Lambda\) into indecomposable sublattices of rank \(1\) and \(2\), and on certain invariants derived from the rank \(1\) components. Parts of the computations have been assisted by a computer. Examples show how these results can be used in many cases to determine the class number within a genus of skew-hermitian lattices of rank \(\geq 2\) over a maximal order in a quaternion division algebra over \(\mathbb{Q}\).