WETZLAR, Germany—Booming smartphone sales are eating into the digital camera market, but the inventor of the 35mm film camera isn’t blinking.

High-end camera producer Leica Camera AG celebrated its first camera’s 100th anniversary last year with a limited-edition model emblematic of its effort to bridge old and new: a digital camera with no display—just an old-style viewfinder. The retro-chic design, which sold for €15,000 ($16,545), was a hit.

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