If everybody is hoarding cash, then, if we assume markets control interest rates and not a central bank, would banks not lower their interest rates as the suppl

If everybody is hoarding cash, then, if we assume markets control interest rates and not a central bank, would banks not lower their interest rates as the supply of deposits increases? Loaning money would become cheaper, and at the same time withdrawing to invest in other things would become incentivized. In sum, high savings would signal that it's safe to start lending and investing.