SEOUL, South Korea — She is facing impeachment and prosecution over allegations of corruption and influence-peddling. One of her advisers is being likened to Rasputin by a shrill South Korean news media. Increasingly large crowds of protesters have taken to the streets, demanding her resignation.

President Park Geun-hye has been paralyzed by a bizarre scandal and an escalating public backlash that could make her the first South Korean leader to be removed from office since her father, the military dictator Park Chung-hee, was assassinated in 1979.

But even as her approval rating slips into the low single digits, Ms. Park has been defiant, meaning that South Korea’s worst political crisis in decades is likely to drag on for months, leaving her conservative government distracted and in disarray while it grapples with a slowing economy and rising household debt.

Moreover, with reports that the cold conflict over North Korea’s nuclear missile program may be heating up as President-elect Donald J. Trump prepares to take office in Washington, the standoff in Seoul could leave the United States with a seriously hobbled ally.