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Strange-metal phenomena often develop at the border of antiferromagnetic order in strongly correlated metals1. Previous work established that they can originate from the fluctuations anchored by the quantum-critical point associated with a continuous quantum phase transition out of the antiferromagnetic order2–4. What is still unclear is how these phenomena can be associated with a potential new phase of matter at zero temperature. Here, we show that magnetic frustration of the 4f local moments in the distorted kagome intermetallic compound cerium palladium aluminium gives rise to such a paramagnetic quantum-critical phase. Our discovery motivates a design principle for strongly correlated metallic states with unconventional excitations. © 2019, The Author(s), under exclusive licence to Springer Nature Limited.