We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4-ε, which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.