KANATA, Ont.—BlackBerry Ltd. is betting its future on a business that makes software for next-generation driverless cars.

Having abandoned production of its once-ubiquitous smartphones, the Canadian company recently refocused its mission around QNX, a core unit whose technology provides a foundation for car-entertainment and information systems. BlackBerry aims to double its QNX engineering staff to about 1,000 in the coming years and spend $76 million to create a center for self-driving car technology near Ottawa.

BlackBerry...