  Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the
Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the
sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map
from a subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ can be extended to a
$CL$-Lipschitz mapping on $\mathbb{R}^m$. In this paper, we construct Sobolev
extensions of such Lipschitz mappings with no restriction on the dimension $m$.
We prove that any Lipschitz mapping from a compact subset of $\mathbb{R}^m$
into $\mathbb{H}^n$ may be extended to a Sobolev mapping on any bounded domain
containing the set. More generally, we prove this result in the case of
mappings into any Lipschitz $(n-1)$-connected metric space.
