We don't do the "anchor baby" thing in Australia.
However, there are certain things that you can expect to happen.
(I will assume in this post that the applicant is lawfully in Australia,
and is not in one of the concentration camps).
First things first - Child Support.
Yes, the father will have to pay Child Support.
And this will have to happen no matter what other arrangements exist
about who lives where and who is with whom in terms of relationships.
His willingness or otherwise to pay is irrelevant - he has to pay anyway.
And it can't come with conditions, such as "I'll pay as long as you don't tell anyone".
Next - Citizenship for the child.
Assuming that the father can be identified, and actually is an Australian citizen himself,
then it is quite likely that the child will be an Australian Citizen by descent.
That is, if the father is a citizen, then the child could well be Australian from birth,
just because the father is (no matter if it actually born in Australia, or elsewhere).
The child may also acquire citizenship of the mother's country of birth
(that depends on the law of that country).
Third - Let's talk about what we are really talking about.
->Can Mum stay automatically because she has an Australian Citizen child?
Automatically? No.
Pregnancy is not a factor that the department regards as beyond the control of the applicant.
So, for example, it would not be a ground on which to waive a "No Further Stay" condition on a visa.
Regardless, your friend should inform the department immediately as this is
a material change of circumstances in her application.
No, they won't kick her out or lock her up just because she's pregnant.
But she does have a duty to make continuous disclosure of her circumstances, and any changes..
Finally - this is a complex subject, but not exactly the first time it's ever happened.
While the law is clear, it is complex, and every case is factually different in some way or other.
This means that and what somebody tells you happened to their next door neighbour's cousin
"about two years ago" is irrelevant.
Your friend needs to engage the services of a lawyer who is
also a Registered Migration Agent, and should do so without further delay.