Automatic addition of date on cell edition (Without script)

I am working on making it possible to see the date when a cell has been edited on a sheet, but I can only think of NOW and TODAY formulas which change every time the sheet is recalculated. I cannot use scripts as my company blocked their use.

Any idea if it is do-able?

Here is what I did for now but cannot think of a way for the date to stay when the data has been written in "B" Column.
Here is the formula I added in C2 =ArrayFormula(IF(ISBLANK(B2:B),"",today()))
It works but the date will change.

Not an ideal way(as done by scripts) but this should serve the purpose in your use-case here. Try this out & see if its feasible for ya:

=arrayformula(if(A2:A,if(B2:B="",now(),B2:B),))

• Enable iterative calculation in sheets setting as in screenshot
• Format the Column B to timestamp format