Thanks Vinny - the appeal cases are particularly interesting and useful. I feel quite hopeful given how essentially straightforward my situation is, when others have been successful in much more complicated situations. The question is whether I will get to the point where I need to appeal, or reapply - neither of which would be very good. A question about appeals (although probably best in another thread) - I understand this can't involve 'new evidence'. So, say they rejected the visa on the grounds that they weren't convinced I had sole responsibility, would I 'appeal' this or have to reapply with further evidence?
Anyway, I found some heartening information last night. In NZ, the mother is legally the sole guardian of the child if she was not married or living with the father at the time of the child's birth (as in my case). So, I am legally her sole guardian and have sole responsibility for all major decisions in her life. As the appeal cases suggest, though, considerations are based on more than the legal status of responsibility and are quite 'practical'.
So here's what I'm going to do - [*apologies for the length of it - it helps me to organise what I'm going to say and do]
1. Write a statutory declaration signed by a notary to say that:
a) My daughter's partner and I were never married or living together and our relationship ended before I was aware of the pregnancy
b) My daughter has always lived with me
c) I have been the sole financial provider for my daughter and her father has never provided any financial support other than irregular child support payments when required by NZ and Australian law
d) I have been the sole source of emotional support and day-to-day caring duties throughout my daughter's life.
e) I have solely made all major decisions relating to my daughter, including where she lives, which schools she attends, and what medical treatment she receives.
f) Her father has had little or no involvement in her life and shows no interest in having a relationship with her. He has seen her once in the past ten years, and often does not reply to emails sent to him by my daughter.
I will also write a letter stating that under NZ law I have sole guardianship of my daughter (and outlining the nature of my 'non'-relationship with her father). I will also include details of his displays of not wanting any involvement. This will include for example his failure to even come straight to the hospital when my daughter was in intensive care when she was 1 and at risk of dying (not arriving until the next day and spending no more than an hour there); and more mundane things like a lack of birthday or christmas gifts; not replying to my daughter's emails and so forth.
And, as mentioned, copies of school fee invoices, medical invoices, etc all showing that I paid for these.
And then just fingers crossed I suppose.
Thank you all for your support and knowledge and advice. It means the world to me. And I'm happy to know I'm welcome here!