Assuming we have a set of points $X=\{x_1,..,x_n\}$, all in $\mathbb{R}^d$, and construct the Vietoris-Rips-Complex $V_\epsilon (X)$ for some distance parameter $\epsilon > 0$. Is it possible to have non-trivial (simplicial) homology groups $H_k(V_\epsilon (X))$ in degree $k\geq d$?

If no, is there a proof? If yes, are there examples, e.g. for $d=2$ with the Euclidean distance?

Update: j.c.'s answer shows that the answer is no for $d=2$ and the $\mathcal{l}_1$- or $\mathcal{l}_\infty$-distances.