WASHINGTON—The fault lines within the Republican Party cracked further on Tuesday as feuding between President Donald Trump and senators intensified within the U.S. Capitol, and anti-establishment activists claimed political momentum outside of it.

Arizona Sen. Jeff Flake, in a speech where he announced he wouldn’t seek re-election, sharply criticized Mr. Trump, declaring himself unwilling to follow the lead of a president whose behavior in office is “not normal” and “dangerous to a democracy.”

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