Yes, injected flu vaccines do not contain live virus, and pose no risk to you or your child if you are breastfeeding. The flu jab is safe for both mother and child.
In fact, women who get the flu vaccine while pregnant or breastfeeding develop antibodies against flu that they can share with their infants through their breast milk. Your breast milk is full of protective antibodies and immune-boosting properties. Once your baby reaches 6 months of age, they can have the flu vaccine themselves but until then they are reliant on the protection from their mother’s flu jab.