HARARE, Zimbabwe—President Robert Mugabe stunned Zimbabwe on Sunday with a speech on live television in which he was widely expected to relinquish his 37-year rule but didn’t—extending days of upheaval that have featured tanks rolling through the country’s capital.

Flanked by generals who took control of his government in a military operation five days ago and by the Catholic priest who takes his confessions, the 93-year-old leader haltingly read out a nearly 20-minute speech, at times tripping over words.

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