The interbank market is essential for liquidity allocation but also a major conduit for systemic risk. This paper develops a dynamic network game that models how financial institutions, acting as borrowers, lenders, or intermediaries, form and adjust bilateral exposures while optimizing liquidity decisions subject to balance-sheet and trust. The equilibrium structure of the lending network is characterized through a dynamic game-theoretic analysis based on best response dynamics and fixed-point conditions, showing how strategic complementarities and balance-sheet feedback generate stable configurations. Building on this framework, the authors analyze how exogenous shocks propagate through the network using a control-theoretic formulation. For asymptotically stable equilibria, the authors give an upper bound for the peak deviation following such a shock. In unstable scenarios, the authors establish sufficient conditions for structural stabilizability using policy interventions. Finally, a constructed case study simulates the dynamic generation of the endogenous equilibrium.