We report on muon spin rotation (μSR) studies of the superconducting and magnetic properties of the ternary intermetallic stannide Ca₃Ir₄Sn₁₃. This material has recently been the focus of intense research activity due to a proposed interplay of ferromagnetic spin fluctuations and superconductivity. In the temperature range T = 1.6–200 K, we find that the zero-field muon relaxation rate is very low and does not provide evidence for spin fluctuations on the μSR time scale. The field-induced magnetization cannot be attributed to localized magnetic moments. In particular, our μSR data reveal that the anomaly observed in thermal and transport properties at T* ≈ 38 K is not of magnetic origin. Results for the transverse-field muon relaxation rate at T = 0.02–12 K suggest that superconductivity emerges out of a normal state that is not of a Fermi-liquid type. This is unusual for an electronic system lacking partially filled ƒ-electron shells. The superconducting state is dominated by a nodeless order parameter with a London penetration depth of λ_L = 385(1) nm and the electron-phonon pairing interaction is in the strong-coupling limit.