Sadly, I don't know of one, and can't find one; doesn't mean it doesn't exist, but the help does indicate that there's only two "special" characters considered, which are replacements for the "&" and "<" characters.
This help page has info on all the special bits you can do for a label, but I don't see anything for the equivalent of non-breaking space.
The Maplex labeling engine has a whole bunch of other rules though, including the maximum number of lines it's allowed to use, and what characters to break a line on. While this isn't exactly what you're asking, it might suffice, assuming you can use maplex. See the stacking docs for further info.
And yes, this seems to be a nigh-duplicate of this question. I've got to remember to start looking before answering.