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We report a highly frustrated spin state persisting down to 0.2 K in the monoclinic Ba₄Ir₃O₁₀ with Ir⁴⁺(5d⁵) ions, which consists of Ir₃O₁₂ trimers of face-sharing IrO₆ octahedra and wavy two-dimensional Ir-O sheets. This iridate conspicuously avoids any long-range magnetic order down to 0.2 K despite strong antiferromagnetic interaction with Curie-Weiss temperature -766 K. The corresponding frustration ratio reaches an astonishing value of 3830. This frustrated spin state is further evidenced by a sizable, linear heat capacity, indicating substantial residual entropy even at milli-Kelvin temperatures. It is equally striking that a mere 2% nonmagnetic Sr substitution for Ba readily lifts frustration and precipitates a long-range antiferromagnetic order at 130 K, suggesting proximity to a quantum critical point potentially with emergent random spin nucleation.