  We study the properties of possible static, spherically symmetric
configurations in k-essence theories with the Lagrangian functions of the form
$F(X)$, $X \equiv \phi_{,\alpha} \phi^{,\alpha}$. A no-go theorem has been
proved, claiming that a possible black-hole-like Killing horizon of finite
radius cannot exist if the function $F(X)$ is required to have a finite
derivative $dF/dX$. Two exact solutions are obtained for special cases of
k-essence: one for $F(X) =F_0 X^{1/3}$, another for $F(X) = F_0 |X|^{1/2} - 2
\Lambda$, where $F_0$ and $\Lambda$ are constants. Both solutions contain
horizons, are not asymptotically flat, and provide illustrations for the
obtained no-go theorem. The first solution may be interpreted as describing a
black hole in an asymptotically singular space-time, while in the second
solution two horizons of infinite area are connected by a wormhole.
