We present an example in ZFC of a locally compact, scattered Hausdorff
non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question
posed by Orihuela, Troyanski and the author in a study of strictly convex norms
on Banach spaces. In addition, we show that the Banach space of continuous
functions $C_0(D)$ admits a $C^\infty$-smooth bump function.