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We report low-temperature muon spin relaxation/rotation (μSR) measurements on single crystals of the actinide superconductor UTe₂. Below 5 K, we observe a continuous slowing down of magnetic fluctuations that persists into the superconducting state, but find no evidence of long-range or local magnetic order down to millikelvin temperatures. The temperature dependence of the dynamic relaxation rate is consistent with spin fluctuations for a three-dimensional weak itinerant ferromagnetic metal, and suggests UTe₂ is close to a ferromagnetic quantum critical point. Our μSR results also indicate that the superconductivity coexists with the magnetic fluctuations and similar to other actinide-based superconductors, UTe₂ has a very low superfluid density.