There are two different challenges here - keeping cool while in camp, and keeping cool while out and about.
In-camp solution are most amenable to serious cooling, and there's no shame in staying in camp and staying cool/sleeping during the heat of the day, then going out during cooler times of the day or night. The simplest solution there is an RV with AC, but that's also the most expensive.
Alternately, you can take a small cargo trailer or box truck, build a temporary wall (and if needed, a door) across the back using plywood or rigid foam insulation, and cool with with a window AC unit and a generator. Still not cheap, but a much smaller investment than an RV. A smaller cargo trailer or truck is low enough in volume that even a small window unit will be very effective. Works much better if you can shade the vehicle too.
If you're not using a vehicle as your base, you're likely in either a hexayurt (not sure if those were common yet in 2009) or a tent. The hexayurt will be much easier to keep cool and will stay that way longer. A tent will require shade over it (with airflow between tent and shade material).
Neither will get you below the ambient temperature without some sort of additional active cooling, but Figjam's battery-powered swamp cooler designs, if made and used properly, can shave another 10 degrees or so off of ambient in most years. Mine did struggle a bit in 2017, but that year was not only really hot, but also more humid than usual.
One mostly-stationary option is a battery-powered misting fan (such as Ryobi's) that sit on a 5-gallon bucket of water, but this requires either multiple battery packs or a way to recharge them, and goes through quite a bit of water - figure a gallon an hour or so.
Alternately, if you're willing to make the trips to Arctica, ice water poured over the head (or into a shallow tub for soaking feet) can be wonderful. You don't even have to use ice water to rinse someone's head, though I'd avoid water that's been sitting in direct sun all day. Dealing with the greywater is the bigger issue there.
For cooling while mobile, you're again probably looking at evaporation solutions.
- handheld spray misters
- neckerchiefs and similar solutions that hold water using agricultural beads and slowly evaporate it, cooling the neck/head. I recommend having extras to rotate out, since after a while the dust and sweat can make them pretty disgusting.
One other possible option - if your wife has a handicapped placard or a government-issued disability ID or license plate number, she could apply for a disability vehicle (DV) license. This would allow you to use a car with air conditioning to move about the playa, so long as she is in the vehicle. More about that here:
https://burningman.org/event/art-perfor ... -vehicles/
