  We introduce $k$-evolution, a relativistic $N$-body code based on
$\textit{gevolution}$, which includes clustering dark energy among its
cosmological components. To describe dark energy, we use the effective field
theory approach. In particular, we focus on $k$-essence with a speed of sound
much smaller than unity but we lay down the basis to extend the code to other
dark energy and modified gravity models. We develop the formalism including
dark energy non-linearities but, as a first step, we implement the equations in
the code after dropping non-linear self-coupling in the $k$-essence field. In
this simplified setup, we compare $k$-evolution simulations with those of
$\texttt{CLASS}$ and $\textit{gevolution}$ 1.2, showing the effect of dark
matter and gravitational non-linearities on the power spectrum of dark matter,
of dark energy and of the gravitational potential. Moreover, we compare
$k$-evolution to Newtonian $N$-body simulations with back-scaled initial
conditions and study how dark energy clustering affects massive halos.
