KEK-BF. We report the first measurement of the lepton forward-backward asymmetry ASYM(FB) for the inclusive electroweak penguin process B --> X(STRANGE) LEPTON+ LEPTON- as a function of Q**2 = M(LEPTON+ LEPTON-)**2, where LEPTON is an electron or a muon and X(STRANGE) is a hadronic recoil system with a strange quark. The results are based on a data sample containing 772*10**6 B BBAR pairs recorded at the UPSILON(4S) resonance, sqrt(s) = 10.56 GeV, with the Belle detector at the KEKB e+ e- collider. For Q**2 > 10.2 GeV**2, ASYM(FB) < 0 is excluded at the 2.3 SIGMA level, where SIGMA is the standard deviation. For Q**2 < 4.3 GeV**2, the result is within 1.8 SIGMA of the theoretical expectation.