Forthcoming time-domain surveys, such as the Rubin Observatory Legacy Survey of Space and Time, will vastly increase samples of supernovae (SNe) and other optical transients, requiring new data-driven techniques to analyse their photometric light curves. Here, we present the ‘Python for Intelligent Supernova-COsmology Light-curve Analysis’ (PISCOLA), an open source data-driven light-curve fitter using Gaussian Processes that can estimate rest-frame light curves of transients without the need for an underlying light-curve template. We test PISCOLA on large-scale simulations of type Ia SNe (SNe Ia) to validate its performance, and show it successfully retrieves rest-frame peak magnitudes for average survey cadences of up to 7 days. We also compare to the existing SN Ia light-curve fitter SALT2 on real data, and find only small (but significant) disagreements for different light-curve parameters. As a proof-of-concept of an application of PISCOLA, we decomposed and analysed the PISCOLA rest-frame light-curves of SNe Ia from the Pantheon SN Ia sample with Non-Negative Matrix Factorization. Our new parametrization provides a similar performance to existing light-curve fitters such as SALT2. We further derived a SN Ia colour law from PISCOLA fits over ~3500 to 7000 A, and find agreement with the SALT2 colour law and with reddening laws with total-to-selective extinction ratio $R_V < 3.1$.