Breastfeed, breastfed, breastfeeding, etc. are sometimes spelled as two words (breast feed, etc.) or hyphenated (breast-feed, etc.), but they are increasingly spelled as one, unhyphenated word, especially outside the U.S. Some publications will continue to resist the trend (new compounds always face resistance for a while), but the one-word forms will likely prevail in the long run.
Exactly how new are the one-word forms? According to the ngram below, which graphs the use of breastfeeding, breast-feeding,and breast feeding in a large number of English-language books, magazines, and journals published between 1950 and 2000, the one-word form prevailed around 1975.
The prevalence of the one-word forms is borne out on the web, where breastfeeding is about three times more common than breast feeding and breast-feeding combined.
There are notable exceptions, though. The New York Times, for one, still favors the hyphenated forms, and such choices in the influential Times tend to ripple through the rest of American publishing, especially in newswriting. The Associated Press stylebook also recommends the hyphenated forms.
Although the one-word, unhyphenated forms are most common throughout the English-speaking world, there is little consistency, and examples of all three spellings (one word with no hyphen, one word with a hyphen, and two words with no hyphen) are easily found: