Refrigeration is hard but I don't think impossible at high ambient temperatures. What immediately comes to mind is compressing gas (which will increase its temperature), letting it equalise with the ambient temperature, then decompressing it, which will lower the temperature (from Charles' Law). No doubt there are many other systems that could be used though.

I'm not an expert on refrigeration at all, but if you're interested, it might be worth checking out the youtube channel Hyperspace Pirate, who does a lot of diy refrigeration systems, and I believe has managed to make liquid nitrogen in his garage using a home made refrigeration system. He also goes very in-depth into the technical details