  Based on a chiral approach to nuclear matter, we calculate the quartic term
in the expansion of the equation of state of isospin-asymmetric nuclear matter.
The contributions to the quartic isospin asymmetry energy $A_4(k_f)$ arising
from $1\pi$-exchange and chiral $2\pi$-exchange in nuclear matter are
calculated analytically together with three-body terms involving virtual
$\Delta(1232)$-isobars. From these interaction terms one obtains at saturation
density $\rho_0 = 0.16\,$fm$^{-3}$ the value $A_4(k_{f0})= 1.5\,$MeV, more than
three times as large as the kinetic energy part. Moreover, iterated
$1\pi$-exchange exhibits components for which the fourth derivative with the
respect to the isospin asymmetry parameter $\delta$ becomes singular at $\delta
=0$. The genuine presence of a non-analytical term $\delta^4 \ln|\delta|$ in
the expansion of the energy per particle of isospin-asymmetric nuclear matter
is demonstrated by evaluating a s-wave contact interaction at second order.
