I would assume that depends on whether OP is the "main permit holder" and she joined him/came here with him based on his permit - or the other way around.

If she's the primary permit-holder then I think it's going to be a problem if she's not truly a resident here any more.

If she's the dependent, how does that work? I don't know the law - can a non-EU dependent permit-holder live apart from the primary permit-holder?

I suppose there's always the "try it and see how it goes" method, but weekly border crossings by a non-EU will probably be noticed at some point and raise questions.

Well my wife has been a dependent on me regarding the permit and residency stuff so my permit status will not change anyway.

She will because she has a family here in Switzerland. By that i mean me as her husband and our daughter...isn't that enough for her to be eligible for holding her B permit?

Depends on whether it is accepted that her focus in live is still in Switzerland when she works in Italy.

Maybe, maybe not. If, for example, she were to leave for her home country again, set up residence and work there, and travel back to Switzerland on the weekends, she would not be entitled to a B permit either. This is pretty much that same situation, except it's even a third country.

Therefore:

This one is key and with someone spending at least 50% in another country, having an apartment/residence of some sort there plus work there, questions will come up about which country is really the "center of her life".

In general, the "center" is assumed to be where the main part of one's personal life is. But with her not really being at that same location, well it'll probably become a judgement call on the end of the official authorities.

Aside from the lack of any permit for Italy, the key problem is her wanting to move there, if only for half the time. If she were to commute daily, it might be slightly easier, at least for the "Swiss side" of things (emphasis on might). But assuming you live in Chur, a daily commute is clearly not an option, hence your situation is considerably more complicated than you apparently thought.

Thanks!
Btw, if she ends up in any types of visa or permit for her study in Italy, will Swiss authorities be notified automatically? If not, are we obliged to notify any authority in Switzerland and how much time we will have for it?
In either case, will she lose her Swiss B immediately, if at all?

Again, she may have to give up her permit, she may not. We cannot tell for sure how this is viewed by the authorities.

Just play it clean as one thing the authorities do not like is someone not adhering to the rules (especially now that the rules are getting tighter).

As for Italy, I´d imagine she needs a permit, how can she work there without one?

Everyone who even changes cantons needs to inform the authorities. So yes, of course if she leaves the country (which is exactly what she would be doing), she is obliged to tell the authorities pretty much immediately. Also because where you live obviously has a number of consequences, think taxes etc.

There's no chance you can "bypass" this by just not telling and hoping no one will find out. I'd generally strongly advise against anything shady. This could have serious consequences not just on her current permit, but may well lead to being banned from both Switzerland and Schengen.

Again, whether she will lose the permit or not no one here can tell. Maybe not even the authorities can right now and will only decide once they have every detail.

And once more, you need to 1) find out how to obtain a work & residence permit for Italy before 2) dealing with the Swiss side of things.

Thanks and much appreciated your time!

We will on Monday contact the Italian embasdy in Switzerland to ask for the precedure. She has a sponsorship letter from university for application of entry visa to Italy. i will post an update here about the progress of the work

Yes, it is. Her focus of life is in Switzerland.

She needs to talk to the Italian authorities to get the appropriate permit in Italy. She will be dual resident with primary residency in Switzerland as her center of life is here.

It's a judgement call but generally center of life is where the spouse and child(ren) are.

Thanks! That is the way we have also thinking so far...

Which is what I said (personal life)

Cross-border commuting imo is out of the question anyway cause
a) they don't live anywhere near the border (a mandatory criterion for the few non-EU cross-border commuters at least in and out of Switzerland - Italy might be more generous, but I doubt it) and
b) she wouldn't return home daily.

And I'd be surprised if a non-EU without permanent (!) and current residency in an EU/Schengen country (again B permit is not permanent residency) could freely take up residency AND work in a number of EU/Schengen countries at the same time. Even less so on a dependent permit in the "main" country.

True we do not live near the border and therefore, daily commuting is no option.

What I do not understand is why she should not have the right for dual residency; in italy cause she studies and in Switzerland cause she has a family. What is more persuading to the authorities for letting my wife hold her permit in Switzerland than the need to treat a 4 year old baby for 3 days a week? Of course she will need to work or study and by chance or whatever it is not in Switzerland.
Btw, i am just thinking bit loudly at this moment writing these lines

Because dual residency doesn't exist for non-EU nationals, you can only be resident in one country.

As Samaire13 said, for a non-EU national living in an EU country to be able to get a cross border permit to work in Switzerland they have to have permanent residency in the EU country. You and your wife do not have permanent residency in Switzerland - that only comes if you are able to get a C permit.

Whether the rules are the same for the EU I don't know which is why you need to speak to both the Italian and Swiss authorities as soon as you can to find out if she's going to be able to take up the position or not. Because there's no guarantee that she will be able to. It may be that the cross border option isn't possible which means the only other possible option (assuming the authorities agree to it) is for her to move permanently to Italy and yes, she would lose her Swiss permit since she would no longer be a resident here.

This is not the simple situation you expected so it's imperative you find out what options, if any, she has.

Medea already answered. You are not EU/CH citizens and therefore do not have the permission to move freely about the EU incl. CH however you please, not for work and/or residence purposes anyway, unless you live somewhere in EU/CH long enough to get permanent residency. At least in Switzerland, even with a C permit there are restrictions on what you can and cannot do and for how long.

If "by chance" her professional future is not in Switzerland, that is too bad, but that doesn't give you any more or less rights. Unless of course you move the family to Italy.