I've deleted some comments that were attempting to answer the question... If you have an answer, please post it as one! Comments are to seek clarification or suggest improvements only. Thanks!

Speed of light in a vacuum is constant, I assume that's what you meant. It's definitely not constant when it moves through different mediums, hence refractive index and things like that.

@SV Indeed, as you've seen, partial answers are not an appropriate use of comments. Yes, people do use them that way sometimes, but they really shouldn't. I guess some people don't know about the rule, some people forget about it, some people know but don't care, some people don't know the difference between the comment and answer text boxes, etc. Anyway, thanks for refraining from answering in the comment section in the future! I hope you'll help us spread the word.

I'm having difficulty understanding the part of the question beginning "delete the earth". You begin by defining speed as the rate of change of distance between the spacecraft and the earth, and then you delete the earth and ask what happened to the speed; well, you're the one who defined speed as rate of change of distance between two objects, so the only person who can answer that question is you; what do you mean by "speed" if there is only one object? What are you really asking here?

Let me clarify what I'm asking for here by way of analogy. Suppose I said "I define 'seniority' as the difference in time between the birth of me and my sister; my seniority is roughly 2.5 years. But what if I didn't have a sister? What would my 'seniority' be then?" If I posed that problem to you, how would you respond? By understanding how you respond to that problem, we can better find an answer to your question.

@EricLippert Following on from your analogy, what if we had a "seniority constant", which we used in a parallel manner to the way we use the speed of light constant (ie extrapolating other constants from it). You could then ask, how is the concept of a constant of seniority possible, if - as you pointed out - seniority is dependent on (in this case) having a sister? Just as seniority is a comparison, ie a concept as opposed to something concrete, and therefore cannot exist in and of itself (let alone be a constant), isn't speed also a comparison?

@Still_learning: You seem to be making a philosophical argument here that again, I don't understand, but yes, speed is a comparison between two things, namely, the rate at which the distance between them is changing. We observe by experiment that this rate has a maximum value, just as we observe by experiment that there is somewhere in the world right now an older brother who is the most years older than his younger sister. But I don't understand the point you're making about "speed cannot exist", so again, it is hard to know how to answer your question; it's very confusing.

Could you provide your definition of speed?

That's what I'm asking you to do, to clarify the part of your question where you ask what happens if you define speed as a relationship between two things and then posit a universe with only one thing in it. That's the part that is confusing me. How I define speed is not particularly relevant.

I'm using the imagery simply to lead up to my actual question. But if you have a problem with my definition of speed then perhaps my question stems from the fact that I have the wrong definition, in which case if I know how you define speed I could then explain better, or perhaps discover that there is actually no question.

@DavidZ Apparently not as the account has been deleted???

@Michael huh, yeah that's weird. :/ There's nothing for the rest of us to do about it now, anyway.

What is your definition of speed? Is it acceleration? Is it velocity? Is it instantaneous velocity? Is it average velocity? Establishing a common understanding of what is being asked might be the place to start first.

"I thought comments could also be used to give a partial answer to the question when you don't have te time to redact, proof-read and format a proper answer." That's the worst way to do it, because comments do not undergo peer review. If you are not sure of your answer and haven't given it due care and attention, then it's even more likely to need that peer review, and you're even more likely to be giving out misinformation. Instead, just leave the answering to someone more ready/capable. :)

Does this answer your question? If all motion is relative, how does light have a finite speed?

This is quite clearly a duplicate of every "please explain relativity to me" question ever, of which there are very, very, very, very, very many. I'm voting to close as a duplicate as the first decent one that appeared in the linked questions.