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We report on the detection of the TRS breaking in the topological superconductor Sr_0.1Bi₂Se₃, probed by zero-field μSR measurements. The TRS breaking provides strong evidence for the existence of a spin-triplet pairing state. The existence of TRS breaking is also verified by longitudinal-field μSR measurements, which negates the possibility of magnetic impurities as the source of TRS breaking. The temperature-dependent superfluid density deduced from transverse-field μSR measurements yields nodeless superconductivity with low superconducting carrier density and penetration depth λ = 1622(134) nm. From the microscopic theory of unconventional pairing, we find that such a fully gapped spin-triplet pairing channel is promoted by the complex interplay between the structural hexagonal warping and higher order Dresselhaus spinorbit-coupling terms. Based on Ginzburg-Landau analysis, we delineate the mixing of singlet- to triplet-pairing symmetry as the chemical potential is tuned far above from the Dirac cone. Our observation of such spontaneous TRS breaking chiral superconductivity on a helical surface state, protected by the TRS invariant bulk topology, can open avenues for interesting research and applications.