This post's title is also the title of a half-page article in "Adhesives and Sealants Industry" (May 2010 issue [*]).
The article is pretty short and not too enlightening, focusing on tests being run at the International Space Station. Adhesive tapes however, have gone much much further in space than just low earth orbit.
The most famous example would be the duct tape that was ubiquitous on the early manned space flights; it was critical to the survival of the Apollo 13 mission, and even used to patch the Lunar Rover.
All of this is small potatoes however to the grand champion of high flying adhesive tape. I wish I could remember the details better, but when I first joined 3M, I worked in one of the tape divisions. Our group received a copy of a letter from NASA stating that portions of an adhesive tape manufactured by our division (I believe it was either 467MP or 468MP) had been used in the construction of the Pioneer spacecrafts and had just recently left the solar system. Now that is a record that won't be beat anytime soon.
[*]Yes, this is a month old. I'm still getting caught up after too much time away from the desk with onsite and offsite client visits.