  We study the effect of the nucleon-nucleon-lambda (NN$\Lambda$) three-body
force on neutron stars. In particular, we consider the NN$\Lambda$ force
recently derived by the J\"{u}lich--Bonn--Munich group within the framework of
chiral effective field theory at next-to-next-to-leading order. This force,
together with realistic nucleon-nucleon, nucleon-nucleon-nucleon and
nucleon-hyperon interactions, is used to calculate the equation of state and
the structure of neutron stars within the many-body non-relativistic
Brueckner-Hartree-Fock approach. Our results show that the inclusion of the
NN$\Lambda$ force leads to an equation of state stiff enough such that the
resulting neutron star maximum mass is compatible with the largest currently
measured ($\sim 2\ M_\odot$) neutron star masses. Using a perturbative
many-body approach we calculate also the separation energy of the $\Lambda$ in
some hypernuclei finding that the agreement with the experimental data improves
for the heavier ones when the effect of the NN$\Lambda$ force is taken into
account.
