DNA extraction kit for microbial mats, rocks, and soil in a remote area

Those are some pretty tough samples to get high-quality DNA from, even in a lab. Your best option is probably to place samples in a nucleic acid preservation buffer and carry them out for extraction later.

That said, there are some potential solutions. Akonni has some kits that don't use any centrifugation and they claim they'll work on soil and stool, so they might be good enough for your applications. They kind of seem like they're based on the same principle as this kit-free protocol (this protocol is for Plasmid DNA, but I'm sure it could be adapted for Genomic DNA with a little R&D on your part). In either case you'd probably want to look into a portable bead-beater to disrupt/lyse your samples. This will shorten your DNA fragments through shearing, but it's probably necessary to get good extractions from any of these materials.

A few companies also make portable centrifuges that run on 12V car plug adapters, assuming you'll have a vehicle somewhere within walking distance. That could open your options up to using a Powersoil kit or even something like a phenol-chloroform extraction, which you might not need a bead-beater for. (Note: you absolutely would need a properly fit-tested, full-face respirator with the appropriate chemical filters to do phenol-chloroform work without a proper fume hood, plus a way to haul out your waste).

If it were me, I'd still probably use a preservation buffer and do the extractions and analysis back at the lab.