Lets compare 'consciousness' of a person with 'turing-machine-ness' of a computer. Neither have anything to do with position in space. Discussion thereoff, is not interesting in my view, the question is if the physical manifestation can create either.

A turing machine is well defined, and cannot be created as far as known, since there is no way to create an infinite tape.

Consciousness is ill defined, but i am sure it exists, because i perceive myself. (Saying perceiving myself is consciousness?) Even if my brain was a system, and one could predict that i will say i perceive myself, does not take away that i do actually perceive myself. This, while the system could make me say 'I perceive myself' without me actually perceiving myself.
It is rather hard to explain what i mean, this post might help The problem you seem to be posing is similar to the problem i pose there, it being the (assumed)uniform behavior of physics. Which seems to imply that consciousnesses should, in the end, be 'one'.
Ilya posted an interesting reply aswel.(read my reply below that link, and that is how far i got on that.)

Edit: I think the question posed in thread is a good idea, although i think "consciousness is limitless in space" is not the clearest way of talking about it.