Beta-delayed proton emission may occur at very low rates in the decays of the light nuclei $^{11}$Be and $^8$B. This paper explores the potential physical significance of such decays, estimates their rates and reports on first attempts to detect them: an experiment at ISOLDE/CERN gives a branching ratio for $^{11}$Be of $(2.5 pm 2.5) cdot 10^{-6}$ and an experiment at JYFL a 95% confidence upper limit of $2.6 cdot 10^{-5}$ for $^8$B.
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