We present the first missing mass spectrum of the unbound nucleus He-10, measured at RIKEN using the Li-11(d,He-3) reaction at 50A MeV. He-10 is believed to be a three-body He-8+n+n resonance beyond the limit of nuclear binding. Our observation of a new decay branch, He-6 + 4n, and of a puzzling reduction of the Li-11(d,He-3)He-10 cross section challenges this view. Moreover, our experiment shows a new trend in the evolution of the proton spectroscopic strength in Li isotopes deduced from the comparison of the (d,He-3) cross sections on Li-7,Li-8,Li-9,Li-11 with theoretical predictions. We discuss new questions about physics beyond the limits of nuclear existence raised by these findings.