I am holding an Australia ETA which is expiring in 23 March 2020. I will be travelling to Australia in February 2020. On the ETA it mentioned that 23 March is the last day I can enter Australia. That means that I will be staying on a expired ETA after that. Is this permitted?
As you have already stated, the ETA needs to be valid on the date that you enter the country. It does NOT need to be valid for the duration of your entire stay.
Thus your ETA that expires in March is fine for an entry in February, even if you stay until after the ETA expires in March.
From the website of the Australian Embassy in London :
My visa will expire when I am in Australia, can I still travel to Australia on it?
The ‘Entry Expiry Date’ of your visa may expire while you are in Australia. The entry expiry date advises the last date you may enter Australia on that visa. Once in Australia you are bound by the ‘Stay For/Until Date’ of your visa. You must depart Australia before the Stay For/Until Date expires. If you fail to depart before this date you will become unlawful in Australia and this may affect future visa applications.