Puppy Linux can be installed but works best running from a USB or disk.
Note: this is different from installing it to usb.I do not like Puppy as a
main OS but some do. It is sparse in terms of what it offers for my taste.
It is meant for occasions when you cannot run a full desktop. Some will
argue that point, but when you can run something like Ubuntu, why would you
want less? It is stripped down to run fast, but it is also not a full
experience. There are trade-offs.

Note: Puppy is NOT based on Ubuntu. (You are asking this in an Ubuntu
group.) There was an Ubuntu based version last spring, but Puppy is its own
distro from scratch. There are different versions kicking around. Slacko the
new one is based on Slackware and Wary is their LTS release, just out.

For installing to usb see: pendrivelinux.com

Running from an ISO is what unetbootin is intended. This is not writable. It
is just like a live CD or DVD. It is read only. You cannot install
applications or save data.

Pendrivelinux.com makes a persistent disk that allows you to boot from the
disk, install and save data. It is like a computer in your pocket.

See http://www.pendrivelinux.com/puppy-linux-on-usb/

This of course depends on your intentions and your needs.