Is it permissible to establish a bank for safekeeping funds and providing liquidity, where the client pays a percentage for the safekeeping service, and the bank uses these funds for its own investments?

There is no objection to the idea of establishing a bank whose function is to preserve money and provide liquidity, provided there is a Sharia supervisory board. If the bank does not use the deposited funds but rather places them in sealed boxes, then there is no objection to taking a fee for this preservation. However, if the bank uses the money in its own investments and guarantees its return upon request, then this is a loan. In this case, it is not permissible for the bank to take a percentage or a lump sum amount for it, unless these fees are in exchange for actual services provided by the bank to the depositor.