Snap Inc., in its first quarterly report as a public company, showed it struggled to maintain strong user growth at its Snapchat vanishing-messaging app, sending shares tumbling and sparking worries about its ability to challenge social-media titan Facebook Inc.

Snap, whose core business relies on selling advertising on the Snapchat messaging platform, reported 166 million daily users in the past quarter, up 8 million from the previous period and up 44 million from a year earlier—its slowest year-over-year growth rate in...