  Let $V$ be a smooth, projective, rationally connected variety, defined over a
number field $k$, and let $Z\subset V$ be a closed subset of codimension at
least two. In this paper, for certain choices of $V$, we prove that the set of
$Z$-integral points is potentially Zariski dense, in the sense that there is a
finite extension $K$ of $k$ such that the set of points $P\in V(K)$ that are
$Z$-integral is Zariski dense in $V$. This gives a positive answer to a
question of Hassett and Tschinkel from 2001.
