Currently, there is no overarching intellectual property law that is enforceable in space. Or strike that, it is sort of in flux. As far as US patent law, there is some enforce-ability if a patent is infringed on a US controlled (35 U.S.C. § 105) objects. But even that has exceptions. Here is a paper on the subject and a more "layman-accessible" article (by same author) on the subject of patents in space. As space gets more commercialized, this area would get more interesting situations for technology development in space.
An area which is being actively researched is 3D printing in space. It used to be that the argument could be made that patent infringement could be alleged for making a component on the ground in a country where the part was made. Now with 3D printing in space, with existing patent law, actors will most likely be able to infringe because there would be no existing "space patent" to infringe. Patents are territorial in which one holding the patent can bar someone else from making, using or selling that infringing component within the territory wherein the patent is issued. With actors in space, there would be no "territory" except the ships or equipment that would be registered under a flag of a country. But as in commercial shipping, most ships that sail on Earth's seas aren't flagged under G-8 nations, but under less "restrictive" flags. The same thing could happen in space. Another matter is for colonization of the Moon and Mars and other celestial objects. Territory definitely does not extend to these objects (Outer Space Treaty). Thus someone printing out objects on the Moon would not be infringing any patents.
As colonizing the Moon and outer space just might be a technologically intensive activity (in which new and useful inventions would be made in situ to solve local problems), some sort of patent regime is needed for space (orbit and cislunar) and outer space.
On one hand, "space patents" would incentivize invention for outer space problem solving, but on the other having no enforceable patents lets those in space make use or sell any invention, which also might be desirable by not disincentiviziing space commercialization. Imagine floating around in your spacesuit and getting a demand-letter from a space-patent troll. :)
I imagine some sort of space patent treaty as part of new space agreements would come into place. It might be another "country" that is designated under a PCT patent filing.
But in the future when the Moon and other colonies get functioning governments, then they would come up with their own territorial patent law such as Lunar Republic patent law, etc.
