Nonequilibrium transport measurements in mesoscopic quasiballistic 2D electron systems show an enhancement in the differential conductance around the Fermi energy. At very low temperatures, such a zero-bias anomaly splits, leading to a suppression of linear transport at low energies. We also observed a scaling of the nonequilibrium characteristics at low energies which resembles electron scattering by two-state systems, addressed in the framework of two-channel Kondo model. Detailed sample-to-sample reproducibility indicates an intrinsic phenomenon in unconfined 2D systems in the low electron-density regime.