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We present the intensity foreground algorithms and model employed within the BeyondPlanck analysis framework. The BeyondPlanck analysis is aimed at integrating component separation and instrumental parameter sampling within a global framework, leading to complete end-to-end error propagation in the $Planck$ Low Frequency Instrument (LFI) data analysis. Given the scope of the BeyondPlanck analysis, a limited set of data is included in the component separation process, leading to foreground parameter degeneracies. In order to properly constrain the Galactic foreground parameters, we improve upon the previous $\texttt{Commander}$ component separation implementation by adding a suite of algorithmic techniques. These algorithms are designed to improve the stability and computational efficiency for weakly constrained posterior distributions. These are: 1) joint foreground spectral parameter and amplitude sampling, building on ideas from Miramare; 2) component-based monopole determination; 3) joint spectral parameter and monopole sampling; and 4) application of informative spatial priors for component amplitude maps. We find that the only spectral parameter with a significant signal-to-noise ratio using the current BeyondPlanck data set is the peak frequency of the anomalous microwave emission component, for which we find $ν_{\mathrm{p}}=25.3\pm0.5$ GHz; all others must be constrained through external priors. Future works will be aimed at integrating many more data sets into this analysis, both map and time-ordered based, thereby gradually eliminating the currently observed degeneracies in a controlled manner with respect to both instrumental systematic effects and astrophysical degeneracies. When this happens, the simple LFI-oriented data model employed in the current work will need to be generalized to account for both a richer astrophysical model and additional instrumental effects.