This study investigates the functioning of modern payment systems through the lens of banks' maturity mismatch practices, and it examines the effects of banks' refusal to roll over short-term interbank liabilities on financial stability. Within an agent-based stock-flow consistent framework, banks can engage in two segments of the interbank market that differ in maturity, overnight and term. We compare two interbank matching scenarios to assess how bank-specific maturity targets, dependent on the dictates of the Net Stable Funding Ratio, impact the dynamics of the interbank market and the effectiveness of conventional monetary policies. The findings reveal that maturity misalignment between deficit and surplus banks compromises the interbank market's efficiency and increases reliance on the central bank's standing facilities. Monetary policy interest-rate steering practices also become less effective. The study also uncovers a dual stability-based configuration in the banking sector, resembling the segmented European interbank structure. This paper suggests that heterogeneous maturity mismatches between surplus and deficit banks may result in asymmetric funding frictions that might precede credit- and sovereign-risk explanations of interbank tensions. Also, a combined examination of macroprudential tools and rollover-based interbank dynamics can enhance our understanding of how regulatory changes impact the stability of heterogeneous banking sectors.