We describe the first-season CO Mapping Array Project (COMAP) analysis pipeline that converts raw detector readouts to calibrated sky maps. This pipeline implements four main steps: gain calibration, filtering, data selection, and mapmaking. Absolute gain calibration relies on a combination of instrumental and astrophysical sources, while relative gain calibration exploits real-time total-power variations. High-efficiency filtering is achieved through spectroscopic common-mode rejection within and across receivers, resulting in nearly uncorrelated white noise within single-frequency channels. Consequently, near-optimal but biased maps are produced by binning the filtered time stream into pixelized maps; the corresponding signal bias transfer function is estimated through simulations. Data selection is performed automatically through a series of goodness-of-fit statistics, including χ 2 and multiscale correlation tests. Applying this pipeline to the first-season COMAP data, we produce a data set with very low levels of correlated noise. We find that one of our two scanning strategies (the Lissajous type) is sensitive to residual instrumental systematics. As a result, we no longer use this type of scan and exclude data taken this way from our Season 1 power spectrum estimates. We perform a careful analysis of our data processing and observing efficiencies and take account of planned improvements to estimate our future performance. Power spectrum results derived from the first-season COMAP maps are presented and discussed in companion papers.