The NA48/2 experiment at CERN reports the first observation of the K± → π0π0μ±ν decay based on a sample of 2437 candidates with 15% background contamination collected in 2003–2004. The decay branching ratio in the kinematic region of the squared dilepton mass above 0.03 GeV2/c4 is measured to be (0.65 ± 0.03) × 10−6. The extrapolation to the full kinematic space, using a specific model, is found to be (3.45 ± 0.16) × 10−6, in agreement with chiral perturbation theory predictions.