Non-hardening adhesive
I think you could use some types of Butylrubber. If I'm not wrong from Isobutylene is derived the Post-it adhesive. However you can find Butyl tape easily in every big hardware store (especially nautical) or online.
EDIT
At this link you can find some more information this is an excerpt:

Tapes and flooring adhesives  Exxon butyl rubber can be used in
adhesive formulations for transparent tape, hot-melt pressure
sensitive adhesives, mastic for pipe wrap tape and vinyl floor tile
adhesives. It can also be used for semiconducting electrical splicing
tape. A rubber splicing tape must be self-fusing; i.e., when wrapped
upon itself, it must fuse to the underlying layer, forming a permanent
bond. The self-fusing is a result of compounding and eliminates the
need for application of a separate adhesive coat. These compounds have
very high tensile strengths and exhibit some degree of recovery or
elasticity. Electrical splicing tapes based on Exxon butyl rubber have
moisture resistance, outstanding aging resistance and good electrical
properties. Polyisobutylene elastomers have long been successfully
used in pressure sensitive adhesives both as primary elastomers and as
tackifiers and modifiers. Exxon chlorobutyl rubber can be used alone
or in blends with other polymers to achieve special properties in
pressure sensitive adhesives.