Is it permissible for employees of a governmental institution to borrow from a bank that the institution stipulated should lend its employees interest-free loans, knowing that the bank benefits from depositing the institution's funds in its interest-based transactions?

If the loan is a Qard Hasan (definition) (a benevolent loan without interest), there is no objection to employees taking it, and the dealings of their institution with an interest-based bank do not harm them, because they themselves have not engaged in usury. The employees are not parties to the forbidden contracts and agreements; rather, the sin falls upon the institution and the bank.