Apple Inc. has held out its fast-growing services unit as a bright spot in an outlook dimmed by soft demand for the iPhone. But the services business faces challenges of its own.

Chief Executive Tim Cook last week underscored that business’s importance in his investor letter warning of slowing iPhone sales, singling out $10.8 billion in services sales for the quarter ended in December. It was the only revenue the company disclosed from its five business segments and another spotlight on a fast-growing business that includes...