We describe the data processing of the Survey on extragALactic magnetiSm with SOFIA (SALSA Legacy Program). This first data release presents 33% (51.34 hr out of 155.7 hr, including overheads) of the total awarded time from 2020 January to 2021 December. Our observations were performed using the newly implemented on-the-fly mapping (OTFMAP) technique in the polarimetric mode. We present the pipeline steps to obtain homogeneously reduced high-level data products of polarimetric maps of galaxies for use in scientific analysis. Our approach has a general design and can be applied to sources smaller than the field of view of the HAWC+ array in any given band. We estimate that the OTFMAP polarimetric mode offers a reduction of observing overheads by a factor 2.34 and an improvement in sensitivity by a factor 1.80 when compared to the same on-source time polarimetric observations using the chopping and nodding mode. The OTFMAP is a significant optimization of the polarimetric mode of HAWC+, as it ultimately reduces the cost of operations of HAWC+/SOFIA by increasing the science collected per hour of observation up to an overall factor of 2.49. The OTFMAP polarimetric mode is the standard observing strategy of SALSA. The results and quantitative analysis of this first data release are presented in Papers IV and V of the series. *The SOFIA Legacy Program for Magnetic Fields in Galaxies (SALSA) provides a software repository at https://github.com/galmagfields/hawc and publicly available data at http://galmagfields.com/.