A hyperbolic phase-field model (PFM) was proposed from the thermodynamic extremal principle for rapid solidification of a binary alloy. In the modeling, not only the interface but also the bulk phases are under non-equilibrium conditions. Dissipation inside the interface, its relations to the sharp interface models, and the previous PFMs are discussed. The solute diffusion in liquid splits into the long-range solute diffusion and the short-range solute redistribution between solid and liquid if the solute diffusion in solid is negligible, being consistent with a recent concept of finite interface dissipation proposed by Steinbach and coauthors (Steinbach in Annu Rev Mater Res 43:89-107, 2013; Steinbach et al. in Acta Mater 60:2689-2701, 2012; Zhang, Steinbach in Acta Mater 60:2702-2710, 2012; Zhang et al. in Acta Mater 61:4155-4168, 2013). Complete solute trapping is predicted when the interface velocity is equal to or larger than the maximal solute diffusion velocity. The interface kinetics is analyzed theoretically and simulated numerically for the rapid solidification of Si-9 at.% As alloy.