The study by Semenova et al. [Phys. Rev. Lett. 117, 014102 (2016)0031-900710.1103/PhysRevLett.117.014102] discovered a type of chimera state known as coherence-resonance chimera (CRC), which combines the effects of coherence resonance (CR) and the spatial property of classical chimeras. In this Letter, we present yet another form of chimera, which we refer to as self-induced-stochastic-resonance breathing chimera (SISR-BC), which differs fundamentally from the CRC in that it combines the mechanism and effects of self-induced stochastic resonance (SISR, previously shown by DeVille et al. [Phys. Rev. E 72, 031105 (2005)1539-375510.1103/PhysRevE.72.031105] to be intrinsically different from CR), the symmetry breaking in the rotational coupling between the slow and fast subsystems of the coupled oscillators, and the property of breathing chimera-a form of chimera state characterized by nonstationary periodic dynamics of coherent-incoherent patterns with a periodically oscillating global order parameter. Unlike other types of chimeras, including CRC, SISR-BC demonstrates remarkable resilience to a relatively wide range of stochastic perturbations and persists even when the purely excitable system is significantly distant from the Hopf bifurcation threshold-thanks to the mechanism of SISR-and globally attracts random distributions of initial conditions. Considering its potential impact on information processing in neuronal networks, SISR-BC could have special significance and applications.