A Dalitz plot analysis of the Ds+ → π ⁻ π ⁺ π ⁺ decay is presented. The analysis is based on proton-proton collision data recorded by the LHCb experiment at a centre-of-mass energy of 8 TeV and corresponding to an integrated luminosity of 1.5 fb⁻¹. The resonant structure of the decay is obtained using a quasi-model-independent partial-wave analysis, in which the π ⁺ π ⁻ S-wave amplitude is parameterised as a generic complex function determined by a fit to the data. The S-wave component is found to be dominant, followed by the contribution from spin-2 resonances and a small contribution from spin-1 resonances. The latter includes the first observation of the Ds+ → ω(782)π ⁺ channel in the Ds+ → π ⁻ π ⁺ π ⁺ decay. The resonant structures of the Ds+ → π ⁻ π ⁺ π ⁺ and D ⁺ → π ⁻ π ⁺ π ⁺ decays are compared, providing information about the mechanisms for the hadron formation in these decays. [Figure not available: see fulltext.].