Mine never gets dirty as I ensure my hands are clean before pressing it.
The Jaguar parts diagram for the Start/Stop button etc is shown here https://parts.jaguarlandroverclassic.com/parts/index/part/id/152.1082.2484.11582/brand/jaguar/ and it includes the updated part numbers.
The latest part number for the bronze coloured button is T2R42910
The latest part number for the black coloured button is T2R42909
Some owners have had issues with buttons becoming sticky as explained in these links....
https://www.ftypeforums.co.uk/viewtopic.php?t=6103
https://www.ftypeforums.co.uk/viewtopic.php?t=8002
Info about wear on the Start/Stop button and how to replace it....
https://www.ftypeforums.co.uk/viewtopic.php?t=2785
https://www.ftypeforums.co.uk/viewtopic.php?t=8221