Nah.

Live images have the image, and free space. Anything you install while they're on uses that free space, and when you turn them off, they still have an untouched OS partition. The space you used to install things gets wiped, essentially.

But you CAN use that space, Linux works as it normally would, just on a USB. Steam could even download a cloud save and upload after you've played, as long as you don't restart the computer.