Can an undergraduate be advised by a professor who is very far away? Lately I have been developing a good relationship with Prof. Wright.

She has clearly* gone out of her way a few times to help me, having a very pleasant, generous, and kind personality. Prof. Wright is searching for students (in her university of course, undergraduate, Master, or PhD). I really like her research and would love to work in her area. However, there's no researcher working in the field anywhere near** the university I study at.

This brings the reason for this question: I have been thinking about
asking her if she could advise me.


Is this unreasonable?
If not, how unlikely is it for her to accept? (To make this practical: what would you do if you were in her shoes?)
Is there a chance of this ruining our relationship? If yes, what can I do to make sure it won't ruin it?
Finally, if I'm going to ask her to be my advisor, it won't (of course and unfortunately) be in person, which means email. What can I do to enhance my chances here?


A few remarks which are (probably) important:


Of course such a setting should benefit both people! I would like to contribute to her research, in a way that is useful to her too;
My field is in Mathematics, being purely theoretical and not requiring lab/field work;
I am moderately independent when it comes to research. More details (in view of comments/answers on this point):

3a. In terms of learning: I've been learning graduate topics by myself for the past two years (I also learned the prerequisites for them in the same way). In particular, I'm used to approaching new topics and learning them by myself.

3b. In terms of producing: I believe*** I would be capable of producing a review article on recent (&lt;10 years) research on certain topics in my field by myself. On the other hand, I can't produce original research. This means I'm used to academic writing, somewhat used to reading original research papers and slightly used to asking original research questions (i.e. can this work be extended for such and such case, what are the main problems this would entail, etc.), but have zero experience with actually producing new research.
I can't move to (anywhere near) her university for now.
(In view of anon's answer) I really don't mind if this won't count for my degree, or that I will be working unpaid. 
On the other hand, I will have to do coursework next year, and
would like to know if (assuming Prof. Wright accepts me as her student) carrying another project while working with her would be a     problem either to her, or to my local, next year, advisor, provided that I can handle these two projects concurrently. (In this case, I would choose a not project with my local advisor sufficiently undemanding (yet not "pointless") for this to work.)


This question is heavily lacking some specific details. I can't provide those, for Prof. Wright is in the SE network too (and these might identify myself).



*To prevent against any kind of bias I may have in making this judgment, I have explained my situation to a few people, who agreed this is indeed a(n unusually) good relationship.

**Near here means less than 2000 km (probably even more).

***I have been doing something similar recently.
It's probably not impossible, but distance isn't the only impediment. If you want her to be your formal advisor then you may need permission from both universities and they may need to work out some compensation, depending on the rules. This will be especially the case if she needs to formally approve (perhaps sign) your dissertation. 

Distance is a problem of course, but with today's communication options it can be overcome in such a case. But you should also consider the possibility, even the necessity, of working face to face on occasion. One way to do this is through attendance at a conference with an extra day or so to work after or before the conference itself. There are other options, of course, including trying to get a small grant from somewhere to cover travel - either yours or hers. 

But asking is certainly appropriate if you are willing to also work to obtain the necessary permissions and deal with the distance issue. 

If she is to be only a secondary advisor it is a bit simpler, but needs the buy-in from your main advisor at your current place. This is easiest if the two know one another and are comfortable working together. But there may still be permission and compensation issues to deal with. 

Make sure you know at the start everyone's expectations about joint authorship of anything you produce, of course.