What can I do about heat rash?

Weather here has been 30 C+ for the past two weeks. I have no air conditioning, and since there are no openable windows, no way to ever get air conditioning. The heat wave isn’t going to break until Wednesday. The problem is that I am now covered head to toe in heat rash. I look like I have chicken pox. It started on my arms and chest, but now it’s marched all the way down to the backs of my hands and onto my face, and my back is so inflamed with heat rash pimples that leaning back in a chair feels like lying on a bed of nails.

To make matters worse, I can’t even take a shower to cool off, since there’s no shower or bath here. I have to take sponge baths in the sink. I also don’t have a refrigerator, so I don’t have access to ice.

Is there anything I can do about this heat rash?