The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as gov- erning operator. The present treatment of the 2-layer case involves technical re¯nements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.