Battleborn

This is actually very simple but the phrasing is a little off. No one on your team is allowed to use a life. You're allowed to have your health depleted and be sent to the "waiting to be revived" state, just as long as that timer doesn't deplete and a player gets sent back to the respawn checkpoint then you should be good.
You might unlock this without even really trying, just stay diligent in paying attention to your teammates, revive anyone you see down which is marked with a big orange + sign. Hold to revive them when next to their body.
If you're having a hard time with teammates not using a life, you can go play private story by yourself or with friends. While solo takes longer you ensure that no one died as long as you don't die. In private you can also enable "HARDCORE" mode which makes it so you have no extra lives but I don't think that's really necessary.