A spree of attacks on two hotels, a coffee shop and a police station in Burkina Faso carried out at least in part by al Qaeda extremists killed at least 30 people, officials said Saturday, as a sprawling Islamist conflict in West Africa spilled into one of the world’s poorest countries.

French special forces and local troops in the Saharan nation spent Friday night and Saturday morning edging their way into the Splendid Hotel, which is located in the capital, Ouagadougou, and is popular with business travelers. At least four...