Inferring high-fidelity constraints on the spatial curvature parameter, ΩK, under as few assumptions as possible, is of fundamental importance in cosmology. We propose a method to non-parametrically infer ΩK from late-Universe probes alone. Using Gaussian processes (GPs) to reconstruct the expansion history, we combine cosmic chronometers (CCs) and type Ia supernovae (SNe Ia) data to infer constraints on curvature, marginalized over the expansion history, calibration of the CC and SNe Ia data, and the GP hyper-parameters. The obtained constraints on ΩK are free from parametric model assumptions for the expansion history and are insensitive to the overall calibration of both the CC and SNe Ia data (being sensitive only to relative distances and expansion rates). Applying this method to Pantheon SNe Ia and the latest compilation of CCs, we find ΩK = −0.03 ± 0.26, consistent with spatial flatness at the O(10−1) level, and independent of any early-Universe probes. Applying our methodology to future baryon acoustic oscillations and SNe Ia data from upcoming Stage IV surveys, we forecast the ability to constrain ΩK at the O(10−2) level.