The woven wool felt used in pianos is probably the nearest to your requirements. Not self-stick, but should take glue without it coming out the other side of the fabric. Very cheap on eBay.

https://www.ebay.co.uk/itm/Piano-Tuning-Wool-Felt-Temperament-Strip-Tapered-Mute-SS/232640896860?epid=1269080637&hash=item362a7a

Ross