We report on measurements of the in-plane magnetic penetration depth (λ_ab) in single crystals of Sr₂RuO₄ down to ≃0.015  K by means of muon-spin rotation-relaxation. The linear temperature dependence of λ⁻²_ab for T ≲ 0.7  K suggests the presence of nodes in the superconducting gap. This statement is further substantiated by observation of the Volovik effect, i.e., the reduction of λ⁻²_ab as a function of the applied magnetic field. The experimental zero-field and zero-temperature value of λ_ab = 124(3)  nm agrees with λ_ab ≃ 130  nm, calculated based on results of electronic structure measurements reported in A. Tamai et al. [High-resolution photoemission on Sr₂RuO₄ reveals correlation-enhanced effective spin-orbit coupling and dominantly local self-energies, Phys. Rev. X 9, 021048 (2019)]. Our analysis reveals that a simple nodal superconducting energy gap, described by the lowest possible harmonic of a gap function, does not capture the dependence of λ⁻²_ab on T, so the higher angular harmonics of the energy gap function need to be introduced.