In a p((-->)e,e'p)pi(0) out-of-plane coincidence experiment at the three-spectrometer setup of the Mainz Microtron MAMI, the beam-helicity asymmetry has been precisely measured around the energy of the Delta(1232) resonance and Q(2) = 0.2(GeV/c)(2). The results are in disagreement with three up-to-date model calculations. This is interpreted as a lack of understanding of the nonresonant background, which in dynamical models is related to the pion cloud.