I like this question a lot! It shows how prejudiced and short sighted we all tend to be and how far financial risk management has to go to cover all bases. Furthermore it is a nice exercise in reconciling market theory with market practice.
A few years (say 10) ago most people considered a floor at r>=0 as being rather obvious. Textbooks stated the possibility of negative rates as major drawback of Gaussian short rate models. This was based on "obvious" arbitrage arguments as the OP made. But practice has shown that this kind of arbitrage is not possible, it is based on lazy thinking.
To reconcile what is going on with theory, imagine a mid-sized institutional investor with 1bn in interest rate sensitive investment. Bank deposits are a no go because banks will refuse the investment since they are charged with negative rates as well. So cash is the obvious way out. But 1bn in cash raises some serious practical questions. How do you transport it? Where do you store it? What about fire and vermin? Is there insurance? All this is not clear but there is one certain thing: Cash is expensive. Swiss pension funds (being charged up to -75bps!) seriously considered this move. Rumor has it that a few larger ones had already scouted out decommissioned army bunkers in the Swiss Alps for storage. (See attached article in German). But this never materialised, which shows that the cost of holding lots of cash is larger than 75bps. I estimate this cost somewhere between 1% to 2%, depending on a lot of factors, such as the size of the operation.
So does this establish a floor at say 2% by arbitrage? No, this would again be lazy thinking. Such a move would create a counter reaction from the central bank. It would either outright refuse to give you the cash (there have been legal opinions in what jurisdictions this is possible or not), issue small bills only or - most radical, most effective - outlaw cash and move to electronic money only.
Well, does this show there is no floor? You guessed right: No, you can't conclude there is a floor. Because faced with outrageous negative rates (for example -10%), investors would consider alternatives, even though they would not match their investment profile. For example, they might start buying gold. Then again, government can restrict your ability to purchase and own gold, then those funds would move to copper and so and so on.
Lesson learned: When stuff hits the fan, a lot can and will happen, that seemed impossible before.