“Young mother nearly dies from a low-carb diet”
in Potential LCHF problems 
Can it potentially be life-threatening to breastfeed on a strict low-carb diet?
While this seems to be exceedingly rare – four published cases ever, all of which ended well – it’s apparently possible to develop ketoacidosis while breastfeeding on a strict low-carb diet. The same thing can happen due to starvation.
Don’t do a strict low-carb diet when breastfeeding – and make sure to eat enough in general.
Correction
Note that the ending of the MailOnline article is misleading. There’s normally no need to “see a doctor immediately” if the breath starts smelling of acetone on a low-carb diet. That’s a common sign of normal ketosis, which is completely safe under normal circumstances (unless you’re a type 1 diabetic, then it signals that you need more insulin).
When breastfeeding however, it’s probably wise not to eat so strict low-carb that you get that smell. Let the weight loss take some more time and go for a more moderate, non-ketogenic, low-carb diet while breastfeeding (50+ grams of carbs per day). And make sure you eat enough. It’s hardly worth taking any health risk at all for losing weight faster.
More
I’ve just recently written more extensively about the rare risk of ketoacidosis when breastfeeding on a strict low-carb diet: