Can HIV be transmitted via breastfeeding?
Yes, it is possible and the chances are pretty high. According to the WHO paper linked below, the rate of partially or completely breastfed children which get infected by this route is about 20-35%, while the rate of children which are not infected and doesn't get breatfed is zero (no surprise here). 

The mechanism by which this infection occurs is currently unknown, but it is hypothesized that the virus enters via the mucosa. See these two articles below for more information:


A review of HIV transmission through breastfeeding.
HIV transmission through breastfeeding : a review.