  We analyze asymptotic properties of higher-dimensional vacuum spacetimes
admitting a "non-degenerate" geodetic multiple WAND. After imposing a fall-off
condition necessary for asymptotic flatness, we determine the behaviour of the
Weyl tensor as null infinity is approached along the WAND. This demonstrates
that these spacetimes do not "peel-off" and do not contain gravitational
radiation (in contrast to their four-dimensional counterparts). In the
non-twisting case, the uniqueness of the Schwarzschild-Tangherlini metric is
also proven.
