Copper contamination causes minority carrier lifetime degradation in p-type silicon bulk under illumination, leading to considerable efficiency losses in affected solar cells. Although the existence of this phenomenon has been known for almost two decades, ambiguity prevails about the underlying defect mechanism. In Paper I of this two-part contribution, we propose the first comprehensive mathematical model for Cu-related light-induced degradation in p-type silicon (Cu-LID). The model is based on the precipitation of interstitial Cu ions, which is assumed to be kinetically limited by electrostatic repulsion from the growing Cu precipitates. Hence, growth and dissolution rates of individual Cu precipitates are derived from the drift-diffusion equation of interstitial Cu and used in a kinetic precipitation model that is based on chemical rate equations. The kinetic model is interlinked to a Schottky junction model of metallic precipitates in silicon, enabling accurate calculation of the injection-dependent electric field enclosing the precipitates, as well as the precipitate-limited minority carrier lifetime. It is found that a transition from darkness to illuminated conditions can cause an increase in the kinetics of precipitation by five orders of magnitude. Since our approach enables a direct connection between the time evolution of precipitate size-density distribution and minority carrier lifetime degradation under illumination, a procedure for calculating the Cu-LID-related lifetime as a function of illumination time is included at the end of this article. The model verification with experiments is carried out in Paper II of this contribution along with a discussion of the kinetic and energetic aspects of Cu-LID.