The suspected attacker died when a homemade bomb exploded in a taxi outside the hospital, while the injured driver named locally as David Perry escaped before the car caught fire.
The passenger is not believed to have been known to MI5.
Home Secretary Priti Patel said the UK terror threat level had been raised from substantial to severe, meaning an attack is highly likely, because the explosion in Liverpool was the second incident in a month.
Four men have been arrested in the city under the Terrorism Act.

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