Does a foreign woman who is getting divorced have a right to stay in the UK if she has a child here?

A neighbour of mine married a Woman in Pakistan. She came on a spouse visa which expired last November but they are going through a divorce and the British Husband did not extend her Visa and cancelled sponsorship and everything. She got pregnant and had a kid here as that was her plan, she wanted to get Pregnant just to stay in the UK. The British Husband doesnt want anything to do with her but is paying some maintence to the child but will rather have child stay with the Mother because if he says he wants the child or contact with child then that will give her a reason to rinse him and make her stay in the UK as like i said that was her plan from the beggining. Now just because you have a Child born in the UK doesntr necessarily give her rights to stay here does it? doesnt the child go back to Pakistan as Child status is with the Mother?.. Otherwise everybody will just get pregnant to stay in this Country, She also made false allegations of voilence and has lied a lot but it seems courts and home office are supporting here. So called Justice hey!.