Isn't the liquidity trap about real interest rates? Here is a so-called thought experiment. Suppose the inflation rate is negative, the nominal interest rate slightly negative, and the real interest rate positive (call it $r$). I think people would rationally make the same savings decisions as if they were in an economy with a real interest rate of $r$, but positive inflation.

But, the liquidity trap is about the nominal interest rate. It says that the nominal interest rate can't drop below zero. I'm trying to understand why, and I think the question can be framed in terms of what banks will decide to do with bonds they own. When the real interest rate is positive, banks will continue to create money and buy bonds, no matter what the long run nominal interest rate is. If the real interest rate were negative, banks would not buy any more bonds, would get less money when selling their current bonds, and they would start to run out of liquidity (no matter what the nominal interest rate is).
The situation you describe, one with negative nominal rates, only can happen with the use of force or where the instrument acts as an insurance policy.  In the case where the instrument serves as an insurance policy, the implicit premium is that negative rate.

To consider why this is the case, consider two investments.  The first investment provides an $\alpha$ percent rate of return where $-100\%&lt;\alpha&lt;0$.  Now consider holding a bank note that pays a zero percent rate of return.  Bank notes have, at times, paid positive rates of return.  Let us assume that we have perfect foreknowledge of future deflation and we know that both will provide a positive real rate of return.

The bank note would always dominate the negative rate of return.  No one would invest in the security that provided negative nominal returns unless they believed the banknote itself was suspect.  There are two ways this dominates.  First, the note provides a higher real rate of return.  Second, however, the banknote will do a better job at preserving its value even under an assumption of future inflation because it pays a higher nominal rate, which is zero percent.  The banknote is both more valuable and simultaneously less risky.

As to the second part of your thinking, that banks would run out of liquidity, you are in error.  The reverse would happen.  As banks buy fewer bonds, they become less leveraged and intrinsically are more flexible and liquid.  Banks become fragile so that their customers can be flexible.  If there are no offers to borrow from the bank at advantageous rates to the bank, then the bank builds reserves instead.

So the banks should be more liquid.  Of course, this does imply that commerce will become less liquid.  It would imply that access to liquidity would be more costly than normal as banks want nominal returns to at least cover costs plus an economic profit on those costs.

What is an issue is whether or not deflation triggers an exchange between debt and equity investing.  The least risky projects can no longer be financed with debt or equity.  It could be asked, "does deflation make the entire system increasingly risky with the passage of time?"  Are there situations that are unwise for the system, though individually rational for each separate actor?  There is also a question of sectoral transfers.  Does commerce transfer profit from itself to banks to preserve treasury operations and basic liquidity functions?