Most Ainu loanwords in regular use are names for plants/animals indigenous to northern Japan, such as reindeer (トナカイ) and sea otter (ラッコ). These loans are old enough that there are usually kanji that can be used for them:
- 馴鹿 (トナカイ, also read じゅんろく)
- 海獺 or 猟虎 (ラッコ)
However, many plant/animal names are usually written in katakana in everyday use (e.g. カバ for "hippopotamus," though it can be written 河馬). Due to the Ainu words being borrowed so long ago and the fact that many people don't realize they aren't "native" Japanese words, I suspect they are usually written in katakana because of this convention.
I've read in at least a few sources that the word さる "monkey" is from Ainu sar'ush, literally "having a tail." Can't promise it's not a false cognate though. It's treated as a native Japanese word in writing.
Using katakana for loanwords was not common practice until after the Meiji restoration (older loans like コーヒー and ガラス can actually be written in kanji, although they generally aren't; たばこ is still written in hiragana).
Aside from that, many place names especially in northern Japan are derived from Ainu. "Sapporo" from Sat-poro-pet and "Shiretoko" from Sir-etok are two off the top of my head. These are generally written in kanji.