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.
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