We exhibit an exactly solvable example of a SU(2) symmetric Majorana spin liquid phase, in which quenched disorder leads to random-singlet phenomenology of emergent magnetic moments. More precisely, we argue that a strong-disorder fixed point controls the low temperature susceptibility χ(T) of an exactly solvable S=1/2 model on the decorated honeycomb lattice with vacancy and/or bond disorder, leading to χ(T)=C/T+DT^{α(T)-1}, where α(T)→0 slowly as the temperature T→0. The first term is a Curie tail that represents the emergent response of vacancy-induced spin textures spread over many unit cells: it is an intrinsic feature of the site-diluted system, rather than an extraneous effect arising from isolated free spins. The second term, common to both vacancy and bond disorder [with different α(T) in the two cases] is the response of a random singlet phase, familiar from random antiferromagnetic spin chains and the analogous regime in phosphorus-doped silicon (Si:P).