  Let $k$ be a positive integer and let $F$ be a finite unramified extension of
$\mathbb{Q}_2$ with ring of integers $\mathcal{O}_F$. An integral (resp.
classic) quadratic form over $\mathcal{O}_F$ is called $k$-universal (resp.
classically $k$-universal) if it represents all integral (resp. classic)
quadratic forms of dimension $k$. In this paper, we provide a complete
classification of $k$-universal and classically $k$-universal quadratic forms
over $\mathcal{O}_F$. The results are stated in terms of the fundamental
invariants associated to Jordan splittings of quadratic lattices.
