WILMINGTON, Del.—In the race to develop the next wave of drugs that use the immune system to fight cancer, scientists scurry up and down escalators in an old department store here.

Their biotechnology company, Incyte Corp., set up shop in a former John Wanamaker store on the outskirts of Wilmington in 2014. Escalators that once ushered shoppers to home furnishings now take researchers to labs that are among the most closely watched in drug development, as Incyte attempts to develop a new generation of cancer immunotherapies.

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