We describe the correction procedure for Analog-to-Digital Converter (ADC) differential non-linearities (DNL) adopted in the Bayesian end-to-end BEYONDPLANCK analysis framework. This method is nearly identical to that developed for the official Planck Low Frequency Instrument (LFI) Data Processing Center (DPC) analysis, and relies on the binned rms noise profile of each detector data stream. However, rather than building the correction profile directly from the raw rms profile, we first fit a Gaussian to each significant ADC-induced rms decrement, and then derive the corresponding correction model from this smooth model. The main advantage of this approach is that only samples which are significantly affected by ADC DNLs are corrected, as opposed to the DPC approach in which the correction is applied to all samples, filtering out signals not associated with ADC DNLs. The new corrections are only applied to data for which there is a clear detection of the non-linearities, and for which they perform at least comparably with the DPC corrections. Out of a total of 88 LFI data streams (sky and reference load for each of the 44 detectors) we apply the new minimal ADC corrections in 25 cases, and maintain the DPC corrections in 8 cases. All these corrections are applied to 44 or 70 GHz channels, while, as in previous analyses, none of the 30 GHz ADCs show significant evidence of non-linearity. By comparing the BEYONDPLANCK and DPC ADC correction methods, we estimate that the residual ADC uncertainty is about two orders of magnitude below the total noise of both the 44 and 70 GHz channels, and their impact on current cosmological parameter estimation is small. However, we also show that non-idealities in the ADC corrections can generate sharp stripes in the final frequency maps, and these could be important for future joint analyses with the Planck High Frequency Instrument (HFI), Wilkinson Microwave Anisotropy Probe (WMAP), or other datasets. We therefore conclude that, although the existing corrections are adequate for LFI-based cosmological parameter analysis, further work on LFI ADC corrections is still warranted.