  We introduce the concept of Roe C*-algebra for a locally compact groupoid
whose unit space is in general not compact, and that is equipped with an
appropriate coarse structure and Haar system. Using Connes' tangent groupoid
method, we introduce an analytic index for an elliptic differential operator on
a Lie groupoid equipped with additional metric structure, which takes values in
the K-theory of the Roe C*-algebra. We apply our theory to derive a
Lichnerowicz type vanishing result for foliations on open manifolds.
