  In this article, we prove the Riemannian Penrose inequality for
asymptotically flat manifolds with non-compact boundary whose asymptotic region
is modelled on a half-space. Such spaces were initially considered by Almaraz,
Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new
approximation scheme for the weak free boundary inverse mean curvature flow,
introduced by Marquardt in 2012, and establish the monotonicity of a free
boundary version of the Hawking mass. Our result also implies a non-optimal
Penrose inequality for asymptotically flat support surfaces in $\mathbb{R}^3$
and thus sheds some light on a conjecture made by Huisken.
