We measure branching fractions and integrated rate asymmetries for the rare decays B-->K(*)l+l-, where l+l- is either e+e- or micro+micro-, using a sample of 384x10(6) BB events collected with the BABAR detector at the PEP-II e+e- collider. We find no evidence for direct CP or lepton-flavor asymmetries. However, for dilepton masses below the J/psi resonance, we find evidence for unexpectedly large isospin asymmetries in both B-->Kl+l- and B-->K*l+l- which differ, respectively, by 3.2sigma and 2.7sigma, including systematic uncertainties, from the standard model expectations.