Timestamp/Date Formula

You would need to use VBA due to the fact that the TODAY() function is volatile and can change each time the worksheet recalculates.

If you put this code in the worksheet module, it will update B-cells with the current date & time only when the adjacent A-cell is changed.

Option Explicit

Private Sub Worksheet_Change(ByVal Target As Range)

    If Target.Cells.Count > 1 Then Exit Sub
    
    If Not Intersect(Target, Columns("A:A")) Is Nothing Then
        Range("B" & Target.Row) = Now
    End If

End Sub