Hi kardut,
You’re welcome!
A wool or cotton mattress protector certainly would could meet your qualifications (there is wool that is washable). Post #89 here talks about various mattress protectors (including some using polyurethane films, which would be for waterproofing without using plastic – which I know is not a need that you have), and about half way through that post are quite a few links to various styles of wool or cotton mattress protectors.
I hope that helps point you in a good direction.
Phoenix