What excel version? Do you have any code in the workbook? Does the cell still has data validation set when you open the file?
Older versions had problem with data validation dropdown button display for workbooks with linked range images (as camera tool). At least in 2016 this issue was fixed in one of updates. The workaround was:
1) add forms combobox anywhere in the worksheet,
2) right-click it and add whole worksheet as data source,
and data validation arrows appear and work.
If it is not your case and you have DV saved, please upload workbook with disappearing DV after removing sensitive data.
combo