  We study the stability properties of static, spherically symmetric
configurations in k-essence theories with the Lagrangians of the form $F(X)$,
$X \equiv \phi_{,\alpha} \phi^{,\alpha}$. The instability under spherically
symmetric perturbations is proved for two recently obtained exact solutions for
$F(X) =F_0 X^{1/3}$ and for $F(X) = F_0 X^{1/2} - 2 \Lambda$, where $F_0$ and
$\Lambda$ are constants. The first solution describes a black hole in an
asymptotically singular space-time, the second one contains two horizons of
infinite area connected by a wormhole. It is argued that spherically symmetric
k-essence configurations with $n < 1/2$ are generically unstable because the
perturbation equation is not of hyperbolic type.
