We show that the effectiveness of unconventional monetary policy is limited by how banks manage liquidity risk, including credit supply, in response to fragile non-bank funding. For identification, we use granular U.S. administrative deposit account, loan-level, and balance sheet data, in conjunction with variations in quantitative easing (QE) and tightening (QT) and an ex-ante bank-level exposure measure to non-banks. QE mechanically increases bank fragility by triggering large inflows of uninsured deposits from non-bank financial institutions. However, we show that banks more exposed to this fragility actively manage this liquidity risk. On the liability side, more exposed banks offer better rates to insured deposits while cutting uninsured rates; by doing so, they shift away from uninsured to insured deposits. On the asset side, exposed banks move away from reserves into higher-yield liquid assets without changing their liquidity regulatory requirements. Moreover, exposed banks also reduce the supply of contingent credit lines to corporate clients. Firms reliant on more exposed banks experience an aggregate reduction in the amount of liquidity insurance they enjoy against future shocks. Our analysis reveals that the fragility of deposit funding can disrupt the complementarity between deposit-taking and the provision of credit lines. Finally, QT does not reverse the QE-associated higher fragile funding from non-bank financial institutions, while overall reserves are lower; consistently, more exposed banks further cut credit lines to non-financial firms.