In a superconductor, the ratio of the carrier density, n, to its effective mass, m*, is a fundamental property directly reflecting the length scale of the superfluid flow, the London penetration depth, lambda(L). In two-dimensional systems, this ratio n/m* (similar to 1/lambda(2)(L)) determines the effective Fermi temperature, T-F. We report a sharp peak in the x-dependence of lambda(L) at zero temperature in clean samples of BaFe2(As1-xPx)(2) at the optimum composition x = 0.30, where the superconducting transition temperature T-c reaches a maximum of 30 kelvin. This structure may arise from quantum fluctuations associated with a quantum critical point. The ratio of T-c/T-F at x = 0.30 is enhanced, implying a possible crossover toward the Bose-Einstein condensate limit driven by quantum criticality.