Thanks very much for your reply. I've just tried it in safe mode, but the fault is still there. It is also there if I log in as root.
I also tried to delete the following Network pane preference files (I had to turn off System Integrity Protection to delete some of these):
com.apple.networkextension.uuidcache.plist
com.apple.networkextension.plist
com.apple.networkextension.necp.plist
com.apple.networkextension.cache.plist
But this did not fix the issue after rebooting.
By the way this is not the only Network pane bug I am experiencing: if I go to System Prefs > Network > Advanced > Hardware, and try to change the Configure option from "automatically" to "manually" in the pop-up menu, as soon as I select manually, it instantly reverts back to automatically.
And if I go to System Prefs > Network > Advanced > TCP/IP, and try to change the Configure IPv4 pop-up menu, again, if I select any menu option, it instantly reverts back to its original value. So I cannot change the Configure IPv4 settings.
If I go to System Prefs > Network > Location (the pop-up menu at the top of the pane), and using Edit Locations..., add another location, that new location also suffers from the same bugs described above.
In terms of unusual software on my Mac running in the background, there's very little. I run Carbon Copy Cloner, Dropbox, a text expander app called Typinator. Quitting Dropbox and Typinator does not fix the issue.