The NA62 experiment reports an investigation of the K⁺ to π⁺ νν^- mode from a sample of K⁺ decays collected in 2017 at the CERN SPS. The experiment has achieved a single event sensitivity of (0.389 ± 0.024) × 10⁻¹⁰, corresponding to 2.2 events assuming the Standard Model branching ratio of (8.4 ± 1.0) × 10⁻¹¹. Two signal candidates are observed with an expected background of 1.5 events. Combined with the result of a similar analysis conducted by NA62 on a smaller data set recorded in 2016, the collaboration now reports an upper limit of 1.78 × 10⁻¹⁰ for the K⁺ to π⁺ νν^- branching ratio at 90% CL. This, together with the corresponding 68% CL measurement of (0.48^+0.72_−0.48) × 10⁻¹⁰, are currently the most precise results worldwide, and are able to constrain some New Physics models that predict large enhancements still allowed by previous measurements.