Time is specifically bound physical circumstances such as the rotation of the earth around the sun, and an 'allotted' number of heartbeats (it and other parts of the body where out). The former circumstance would be altered in space, but the latter only by technology, but the key here would maintaining the brain, the most complex 'organ' lest susceptible to technological improvement or replacement.
However, you raise an interesting point. 'When you fall asleep, where in the world goes your sense of time?' As many do, I believe there is a 'more real world' beyond this one, whether it be Plato's realm of forms or the Mind of God or something else (not knowing for sure what it is, or if it really exists, and having no 'real world' point of reference, we have to cal it something), it is the 'world' we enter when we dream. This world is bound be neither time nor space. What I conclude from this is that time is a function of physicality, and I believe (as Buddha did) that physicality is a functional illusion.