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Hi. I'm having this problem too, with Excel v. 16.20 and most earlier versions going back to 15.nn. When I first created the sheet a couple years ago, it worked perfectly. Data validation drop down menus worked fine, so I have several sheets in a workbook where I have such data validation. It seems to me I began to have this problem in the last year or so.
Anyway, I have fixed it.
The sheet in question is a multi-column table. Three of the columns have data validation using dropdown menus. That validation method which originally worked perfectly in Excel 2011 and I think in early versions of 15.nn wasn't working (no arrows) in any of the three columns. After I copied everything into a new sheet following other advice in this thread, two of the columns were fixed, the arrow appeared in any selected cell. The third column, which previously also showed no arrow, now showed an arrow which didn't react and produce a drop down menu when clicked. So the arrows were present in the third column but were not functioning.
I looked at the data for the dropdown in the validation dialog. It consists of two items: ✓ (a checkmark) and a blank space. Without going into detail about the reasons for this, the blank space was the culprit. If I remove it from this column's dropdown, the dropdown works correctly. Other fiddling seems to show that, in effect, you cannot use a blank space as a choice in a dropdown menu (even though in the past it worked fine with a blank space).
I substituted a character other than a blank space so that the user could remove a checkmark. A workable but not perfect solution. The third column dropdown now works.
Hope this helps.