As previously mentioned, cling film is a sod to use in this situation, you are better off with a plastic bag. It folds up nice and small if you do it sensibly, has the same function, an if it's a burn to a hand, you just put the whole hand in the bag *AFTER* you've cooled it down.
For cooling it down, consider carrying something like
Burn gel. Tho the old stalwart of water will work just as well, if you have enough of it. And you need a lot. The ideal is to put the affected area under RUNNING water for at least 10 minutes. That's a lot of water...
Also note, you don't have to just use water, any liquid that you can drink is ok for cooling a burn, in a pinch.
J