Can Linux Lite be updated when using persistent live USB version?

[Moltke]

Is is not possible to do updates while using LL in this way?

Yes, it is. You need to edit the sources.list file, and remove these lines cdrom://Linux Lite 5.4 - Release amd64 dists/ InRelease and cdrom://Linux 20.04 - Release amd64 dists/ Release as those are preventing the system to be updated.
1. Launch xfce4-terminal.
2. Type

Code:

sudo nano /etc/apt/sources.list.d/linuxlite.list

press Enter, type in your password when asked, and hit Enter again.
3. Once the file is open to edit, locate and remove the lines cdrom://Linux Lite 5.4 - Release amd64 dists/ InRelease and cdrom://Linux 20.04 - Release amd64 dists/ Release
4. To save your changes, press Ctrl + o and hit Enter
5. To close the editor nano, press Ctrl + x and hit Enter
6. Try updating Linux Lite again, it should work this time.

Keep in mind one thing though, while using any Linux distro in persistent mode, the kernel can't be upgraded, other components can and will, but not the kernel. This is because the kernel needs to make a few modifications to the system, which can't be made while in persistence mode.

Hope this helps!