  We study linearized equations of a ghost-free gravity in four- and
higher-dimensional spacetimes. We consider versions of such a theory where the
nonlocal modification of the $\Box$ operator has the form $\Box
\exp[(-\Box/\mu^2)^N]$, where $N=1$ or $N=2n$. We first obtain the Newtonian
gravitational potential for a point mass for such models and demonstrate that
it is finite and regular in any number of spatial dimensions $d\ge 3$. The
second result of the paper is calculation of the gravitational field of an
ultrarelativistic particle in such theories. And finally, we study a head-on
collision of two ultrarelativistic particles. We formulated conditions of the
apparent horizon formation and showed that there exists a mass gap for
mini-black-hole production in the ghost-free theory of gravity. In the case
when the center-of-mass energy is sufficient for the formation of the apparent
horizon, the latter has two branches, the outer and the inner ones. When the
energy increases the outer horizon tends to the Schwarzschild-Tangherlini
limit, while the inner horizon becomes closer to $r=0$.
