ABSTRACT Doppler spectroscopy is a powerful tool for discovering and characterizing exoplanets. For decades, the standard approach to extracting radial velocities (RVs) has been to cross-correlate observed spectra with a weighted template mask. While still widely used, this approach is known to suffer numerous drawbacks, and so in recent years increasing attention has been paid to developing new and improved ways of extracting RVs. In this proof-of-concept paper, we present a simple yet powerful approach to RV extraction. We use Gaussian processes to model and align all pairs of spectra with each other; we combine the pairwise RVs thus obtained to produce accurate differential stellar RVs, without constructing any template. Doing this on a highly localized basis enables a data-driven approach to identifying and mitigating spectral contamination, even without the input of any prior astrophysical knowledge. We show that a crude implementation of this method applied to an inactive standard star yields RVs with comparable precision to and significantly lower rms variation than RVs from industry-standard pipelines. Though amenable to numerous improvements, even in its basic form presented here our method could facilitate the study of smaller planets around a wider variety of stars than has previously been possible.