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This study investigates the dynamic behavior of a three-body tethered satellite system with a flexible tether, focusing on a scenario in which a payload is transported along the tether connecting two satellites. Traditional models use rigid tethers, limiting the analysis of complex dynamics such as tether deformation, slack, and rebound. To address these limitations, we employed the absolute nodal coordinate formulation (ANCF) to model the flexibility of the tether. We derived nonlinear equations of motion using Lagrange's equation and solved them using the Newmark time integration method to obtain the dynamic responses of the satellite system. Our findings revealed that the Coriolis effect caused significant deviations in the payload trajectory as its mass and speed increased, leading to greater tether deformation and slack, and potential system destabilization. Additionally, axial force fluctuations in the tether varied notably as the payload moved, transitioning between tensile and compressive states. This study provides a more accurate representation of three-body tethered satellite systems by incorporating tether flexibility, offering valuable insights into the dynamic behavior and stability of the system. [ABSTRACT FROM AUTHOR]