  We consider the weighted Radon transforms $R_W$ along hyperplanes in $R^d, \,
d \geq 3$, with strictly positive weights $W = W (x, \theta), \, x \in R^d, \,
\theta \in S^{d-1}$. We construct an example of such a transform with
non-trivial kernel in the space of infinitely smooth compactly supported
functions. In addition, the related weight $W$ is infinitely smooth almost
everywhere and is bounded. Our construction is based on the famous example of
non-uniqueness of J. Boman (1993) for the weighted Radon transforms in $R^2$
and on a recent result of F. Goncharov and R. Novikov (2016).
