Mother of UK born child - overstay case

[sanson]

My friend's case is a bit tricky. Please, if anyone has gone through a similar situation ANY advice would be highly appreciated:

FACTS:
He is British, she is Argentinian. They got married in her home country in 2009, she applied for a UK spouse visa from her home country, got the visa - valid for 2 years, and came to live to the UK. In 2010 they had a baby (born in the UK). In 2011 her visa expired and she didn't do anything about that, she has overstayed in the UK unil now, 2012 (1 year of overstay)

SITUATION NOW:
She now wants to apply for UK citizenship: she took the test, passed, went to the appointment with all the requirements and only THEN was she told by the Home Office that she cannot apply from the UK because she is current living illegally in the UK, that she has to do that from her home country.

Then, based on the fact that she is a mother of UK born child and spouse of British man, she was advised (by the Home Office as well) to appeal against that decision because if she now leaves the UK she could be banned from entering again for 10 years.

QUESTIONS:
1- Is it possible to appeal?
2- How feasible is for her to win the appeal from the UK and get the UK citizenship? How long would this process take?
3- Is it true that because she is a mother of a UK born baby (and still married) she cannot be deported?
4- Is there any other way out of this situation for her?

Thank you in advace for your answers and further comments, she is really having a hard time.

[vinny]

She cannot apply for citizenship. However, she may be able to apply by post for ILR.

[bob-russell]

Worst case scenario, I suspect, would be that she is granted Discretionary Leave To Remain.

Not as 'good' as ILR but it is unlikely that any husband or wife of a British Citizen would be made to leave the country!

A British citizen child added to the equation makes it even less likely! The human rights lawyers would have a field day!