Bitcoin in the far future
The extraterrestrial mining problem is the prospect of Bitcoin mining occurring on bodies other than the Earth.
Lunar mining
Mining bitcoins on the moon is feasible, as transactions and blocks between it and the Earth would only be delayed by a few seconds. The worst that this would cause is the occasional orphaned block when a block is solved on both the Earth and the Moon within this several-second timeframe. Transactions could occur when broadcast from either body and nodes would be able to communicate with other nodes on either body.
This also applies to satellites within the Moon's orbit.
Martian mining
Mining on Mars is a bigger problem, because it would take several minutes for communication from the Earth to reach Mars, depending on their distance at the time of broadcast. As such, it is uncertain that a block solved on Mars would reach the Terrestrial block chain in time for a block to be solved on Earth instead.
There would be windows of low latency when Earth and Mars become close enough for a block to be transmitted from one to the other in less than five minutes, but they would only ever be temporary.
Proposed solutions
One or many sidechains could be located on Mars for use by Martian nodes, miners and transacting parties. This means that a block solved on a Martian sidechain would be able to propagate completely without ever needing to wait to travel to Earth. With a two-way peg to the main Terrestrial chain, bitcoins could be transferred from Earth to Mars in a manner of days (which would probably be faster than shipping people and goods from Earth to Mars anyways) and bitcoins transferred between people on the same planet would be instant as usual (still of course requiring confirmations, which, depending on the sidechain's rules, would usually take ten minutes). In this situation a block solved on Mars would not produce new bitcoins, but only collect Martian transaction fees. However, by the time mining on Mars becomes feasible, the block reward would probably be very low anyway and transaction fees would comprise most of miner's revenue in the first place. As Martian population would probably be much lower than Earth's, Martian difficulty would be as well.
If faster-than-light data transfer can be utilized for transmitting across interplanetary scales, the window of low-latency would be expanded without need for an alternative chain. This window could potentially be expanded to the point of bring reliable 24/7, eliminating any further disadvantages to extraterrestrial Bitcoin at the protocol level.
Extrasolar mining
Where faster-than-light technology is required to reach somewhere in a reasonable amount of time, faster-than-light technology would be required to communicate with anyone there in the first place without significant delay. Whether bitcoin will be able to function under these conditions depends on what kind of technology will exist when they will be reproducible.