LIMA, Peru — For the heads of state mingling in the meeting halls, it was the leader who had been uninvited to the gathering who posed the biggest challenge.

President Nicolás Maduro of Venezuela was the outcast at the Summit of the Americas in Lima, Peru, where leaders came to hash out trade deals and mend political fences.

Unwelcome at this meeting, Mr. Maduro’s isolation has only grown as he rules the country with an increasingly authoritarian hand and presides over the worst economic crisis in generations, with Venezuela’s suffering extending from food shortages to hyperinflation.

For all the leaders here, the threat posed by Venezuela is clear, and for many, it is pressing.

Impoverished Venezuelans have spilled over the borders of Colombia and Brazil, seeking food and medicine, with 500,000 estimated to have arrived in Colombia alone. Despite all these troubles, Mr. Maduro is seeking re-election in May in a vote in which his most popular challengers are banned, imprisoned or have pledged to boycott.