The B (1) Pi-X (1) Sigma(+) band system of the (KLi)-K-39-Li-7 molecule was investigated by Doppler-free polarization spectroscopy in a heat-pipe oven. Precise molecular constants for levels upsilon = 0-3 in the ground X (1) Sigma(+) state and levels upsilon = 0-2 in the excited B (1) Pi state were determined. The e-parity components in all three vibrational levels of the B state were found to be subject to rotational perturbations, assigned to interaction with the C (1) Sigma(+) state. The deperturbation analysis provided preliminary constants of the perturbing C slate and values of the coupling matrix elements.