  We investigate the small-scale static configurations of K-mouflage models
defined by a general function $K(\chi)$ of the kinetic terms. The fifth force
is screened by the nonlinear K-mouflage mechanism if $K'(\chi)$ grows
sufficiently fast for large negative $\chi$. In the general non-spherically
symmetric case, the fifth force is not aligned with the Newtonian force. For
spherically symmetric static matter density profiles, the results depend on the
potential function $W_{-}(y) = y K'(-y^2/2)$, which must be monotonically
increasing to $+\infty$ for $y \geq 0$ to guarantee the existence of a single
solution throughout space for any matter density profile. Small radial
perturbations around these static profiles propagate as traveling waves with a
velocity greater than the speed of light. Starting from vanishing initial
conditions for the scalar field and for a time-dependent matter density
corresponding to the formation of an overdensity, we numerically check that the
scalar field converges to the static solution. If $W_{-}$ is bounded, for
high-density objects there are no static solutions throughout space, but one
can still define a static solution restricted to large radii. Our dynamical
study shows that the scalar field relaxes to this static solution at large
radii, whereas spatial gradients keep growing with time at smaller radii. If
$W_{-}$ is not bounded but non-monotonic, there is an infinite number of
discontinuous static solutions but they are not physical and these models are
not theoretically sound. Such K-mouflage scenarios provide an example of
theories that can appear viable at the cosmological level, for the cosmological
background and perturbative analysis, while being meaningless at a nonlinear
level for small-scale configurations. This shows the importance of small-scale
nonlinear analysis of screening models.
