Can a breastfeeding mother with HIV infect her child?

Yes,for sure.

A mother with HIV virus cannot breast feed her child without infecting the child.

Well am not a medical personnel anyway but I do know that someone with HIV can give birth without the baby carrying the virus due to some drugs available right now but the mother can't breastfeed such child.

HIV is a blood disease and the milk from the breast is also connected to the bloodstream so like I said earlier it's not possible right now medically for it to happen.