One possibility is basalt fiber cloth. It is made on Earth using only basalt rock, which is melted and drawn into threads as fine as 9 microns. A small amount of sizing is applied to the threads to prevent sticking during weaving, that accounts for less than one percent of the final mass. Here, the rock is washed, but that is done to remove organic materials, which don't exist on Mars or the Moon, so that step could be skipped. I have been interested in it for the Moon.
It requires a large furnace that can sustain something above 1200 degrees Celsius. The article linked above mentions 1500 Celsius, but there are significant chemical differences between basalt on Earth and on the Moon that will affect the operating temperature. I don't know what the situation is with Martian basalt, or how it varies over the planet. Only certain basalt deposits are considered suitable on Earth for production of fiber - however, if one was on a dead planet, one could likely find a way to make do with less ideal basalt.
This article from Composites World goes over the process and issues well. Here are a few key quotes:
Unlike glass, which is transparent, the opaque basalt absorbs rather
than transmits infrared energy... the melting basalt must be held in
the reservoir for extended periods of time — up to several hours — to
ensure a homogenous temperature. Basalt producers have employed
several strategies to promote uniform heating, including the immersion
of electrodes in the bath.
Unlike glass, basalt fibers feature no secondary materials. The
process requires only a single feed line to carry crushed basalt rock
into the melt furnace. On the other hand, basalt fiber manufacturers
have less direct control over the purity and consistency of the raw
basalt stone... despite its ready availability from mines and open-air
quarries around the world, only a few dozen locations contain basalt
that has been analyzed and qualified as suitable for manufacture of
continuous thin filaments.
Basaltex, for example, found early on that woven basalt fabrics
straight from a weaver's loom were fragile and easily damaged when
handled, exhibiting broken fibers when sharply folded or bent, and
were irritating to the skin. In order to make the product more stable,
Basaltex developed a proprietary silane-based sizing that facilitates
the post-manufacture processing.
Silane is composed of silicon and hydrogen.
There are many manufacturers and suppliers. I didn't want to single one out, but I do have a sample pack and a few meters of cloth. In terms of clothing, the textile is a bit stiff, but has a silky finish that isn't irritating. I haven't ever had it next to my skin for any length of time, though. For any application other than clothing, it should do the job of any textile or rope just fine.