In this paper, we derive sharp nonlinear dimension-free Brascamp-Lieb inequalities (including hypercontractivity inequalities) for distributions on Polish spaces, which strengthen the classic Brascamp-Lieb inequalities. Applications include the generalization of Mr. and Mrs. Gerber's lemmas to R\'enyi entropies and distributions on Polish spaces, the strengthening of small-set expansion theorems, and the characterization of the exponent of $q$-stability of Boolean functions. Our proofs in this paper are based on information-theoretic techniques and coupling techniques.