  In this paper, we study the effective field theory (EFT) of dark energy for
the $k$-essence model beyond linear order. Using particle-mesh $N$-body
simulations that consistently solve the dark energy evolution on a grid, we
find that the next-to-leading order in the EFT expansion, which comprises the
terms of the equations of motion that are quadratic in the field variables,
gives rise to a new instability in the regime of low speed of sound (high Mach
number). We rule out the possibility of a numerical artefact by considering
simplified cases in spherically and plane symmetric situations analytically. If
the speed of sound vanishes exactly, the non-linear instability makes the
evolution singular in finite time, signalling a breakdown of the EFT framework.
The case of finite (but small) speed of sound is subtle, and the local
singularity could be replaced by some other type of behaviour with strong
non-linearities. While an ultraviolet completion may cure the problem in
principle, there is no reason why this should be the case in general. As a
result, for a large range of the effective speed of sound $c_s$, a linear
treatment is not adequate.
