I was too curious to wait for you to check if it works. So I tried and here is my result:
Yes, it works with the following recipe :-)
Create a live-only drive by cloning ubuntu-16.04.1-desktop-amd64.iso to a 32 GB USB 3 pendrive with mkusb
Add the boot option toram and boot
When running live, unmount all mounted partitions in the pendrive (in this case it was only one), run in a terminal window
sudo umount /media/{and press the TAB key for 'tab completion'}
# or
sudo umount /media/*/* # if more than one partition
Check that all partitions on the pendrive are unmounted with
df
Start the installer with the desktop icon
Follow the instructions like any other installation
Reboot
I did it in a laptop computer with 4 GiB RAM in BIOS mode without any internal drive.
If there is an internal drive, things may be more tricky in order to avoid writing bootloading things to the internal drive, but this is not due to the fact that it is installed to the same drive as the system was booted from.
-o-
Edit 1: I tested in UEFI mode. The final 'Reboot' got stuck at the plymouth 'five dots'.
and when rebooted, the installed system works. A better alternative is to
and it finishes and reboots beautifully.
-o-
Edit 2: You find help how to add the boot option toram at this link,
How to unmount a live DVD/USB?