**Core Concept**
Hyperbilirubinemia in children can be caused by various factors, including genetic disorders, liver diseases, and conditions affecting bilirubin production or conjugation. The question requires the identification of a condition associated with hyperbilirubinemia.

**Why the Correct Answer is Right**
Fanconi's syndrome is a rare genetic disorder characterized by impaired renal tubular function, leading to excessive loss of essential nutrients, including bilirubin conjugates. This results in an inability to properly conjugate and excrete bilirubin, causing hyperbilirubinemia. Fanconi's syndrome affects the proximal renal tubules, disrupting the reabsorption of glucose, amino acids, and other substances, including bilirubin conjugates.

**Why Each Wrong Option is Incorrect**
**Option A:** Breast milk jaundice is a condition that occurs in newborns, causing hyperbilirubinemia due to substances in the breast milk that increase bilirubin reabsorption from the intestines. However, it is not related to Fanconi's syndrome.

**Option B:** Cystic fibrosis is a genetic disorder affecting the respiratory, digestive, and reproductive systems, primarily due to mutations in the CFTR gene. It does not directly cause hyperbilirubinemia.

**Option D:** α-1 antitrypsin deficiency is a genetic disorder that affects the liver and lungs, but it is primarily associated with liver disease and emphysema, not directly with hyperbilirubinemia.

**Clinical Pearl / High-Yield Fact**
Fanconi's syndrome is a rare condition, but it is essential to consider it in the differential diagnosis of hyperbilirubinemia, especially in patients with a history of renal disease or tubular dysfunction.

**Correct Answer: D. α–1 antitrypsin deficiency**