The deadlock will continue into a fourth day, since no business can go ahead until a speaker is picked.
Lawmakers in the sharply divided chamber will reconvene on Friday, the second anniversary of a riot by Trump supporters at the US Capitol.
Rules do not require the speaker to be a member of the House, and on Thursday, Florida Republican rebel Matt Gaetz cast a protest ballot for Mr Trump to serve in the role. Colorado Republican Lauren Boebert nominated a lawmaker from Oklahoma, telling her colleagues to move past Mr McCarthy.
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