As it became clear, early Friday morning, that Britons had voted to leave the European Union, a far-right nationalist politician, Nigel Farage, told cheering supporters that the goal of his once-fringe United Kingdom Independence Party had finally been met.

“Dare to dream that the dawn is breaking on an independent United Kingdom,” Farage declared. The victory, he added, was one for “honesty, decency and belief in nation,” achieved “without a single bullet being fired.”

That last claim struck a nerve with those for whom the defining event of the referendum campaign was the horrifying assassination, with three bullets, of Jo Cox, an internationalist, pro-European legislator whose killer shouted “Britain first” and “Keep Britain independent” as he shot and stabbed her to death.

Farage and his colleagues were quickly congratulated by the leaders of nationalist, far-right parties in the Netherlands and France, Geert Wilders and Marine Le Pen, who both called for similar referendums in their countries.

Closer to home, however, another part of Farage’s statement seemed at odds with the mood in some parts of the country. Namely, his confident assertion that citizens of the United Kingdom comprise a single nation. As the results were tallied, it became obvious that there was a clear disparity of outcomes across the kingdom’s four nations: While England and Wales voted to leave the EU, Scotland and Northern Ireland both voted to stay in it, which raised the possibility that the decision to withdraw from one union could trigger the imminent collapse of another.

To start with, as many observers in Scotland noted, their nation, which rejected independence in another referendum just two years ago, partly out of a desire to remain in the EU, voted overwhelmingly against the decision to leave. That led to widespread speculation that the Scottish National Party, which rules the local government, could now demand a second referendum on independence soon.

Hours later, Nicola Sturgeon, Scotland’s first minister, told reporters that she would do everything she could to make sure that her nation’s vote to be part of the EU was honored, and “an independence referendum is now highly likely.”

Sturgeon also mentioned that she had spoken with the leader of another part of the U.K. that voted overwhelmingly to remain in the EU, Sadiq Kahn, the mayor of London, and they promised to work together.

That statement prompted a wave of jokes about London declaring its independence too, but also some serious suggestions that the pro-European forces might still fight to keep the U.K. in the EU in some form.

As if any more fuel was needed to add to this fire, Donald Trump arrived in Scotland on Friday morning to promote one of his golf courses, and told waiting reporters that the British exit from Europe was “a great thing.”

Trump, whose mother was from a remote part of Scotland (the Western Isles, where 55 percent of voters opposed leaving the EU), seemed oblivious to nationalist sentiment there, telling reporters the vote meant, “Basically, they took back their country.”

He also displayed little sympathy for fears about the country’s economy prompted by the sharp drop in the value of the pound overnight, once the direction of the vote became clear — arguing that it would be good for business at his golf course.

(Later in the day, video of Trump’s rambling remarks on his golf course as the global economy shuddered over Brexit became fodder for a Hillary Clinton ad mocking him.)

Meanwhile in Northern Ireland, which also voted against leaving the EU by a clear majority, the nationalist Sinn Fein party called for an immediate referendum on unification with the Republic of Ireland, which will remain in the economic union.

“This outcome tonight dramatically changes the political landscape here in the north of Ireland and we will be intensifying our case for the calling of a border poll” on a united Ireland, the party’s chairman, Declan Kearney, said on Friday morning.

“The British government as a direct result have forfeited any mandate to represent the interests of people here in the north of Ireland in circumstances where the north is dragged out of Europe as a result of a vote to leave,” he added.

If the United Kingdom does leave the EU — after what was, legally, just an advisory referendum — Northern Ireland will be the only part of the country to share a land border with the economic bloc, reintroducing the need for immigration and customs controls with the Republic of Ireland that were phased out in recent years.

Sinn Fein was a party to the Good Friday agreement of 1998, which ended decades of violence by devolving power to a regional assembly and acknowledging that the two parts of Ireland could eventually be united. As part of that agreement, the British government’s secretary of state for Northern Ireland is obliged to call a referendum on dissolving the border between the two parts of Ireland once there is clear evidence of public opinion favoring Irish unity.

As Brendan Donnelly, a former member of the European Parliament, wrote recently, “It is not by chance that in the Good Friday agreement … so much emphasis is laid on the membership of both the United Kingdom and the Republic of Ireland in the European Union.” Under that arrangement, the border has become as invisible as one between two American states.

“To install physical checkpoints along the border would instantly undermine a hard-won peace, and the psychological impact alone would be catastrophic,” Kathryn Gaw, a Belfast-based journalist wrote this week. “A return of those barricaded towers and armed checkpoints will stir up emotional memories for many Northern Irish people who witnessed years of violence in border towns such as Newry, Omagh and Derry, and there is a very real fear that they may lead to a resurgence of dissident activity.”

Although the unionist strongholds in Ulster voted to leave the EU on Thursday, the difficulty of living once again with physical barriers between the two parts of the island could make it seem more attractive to be part of a European Ireland than an inward-looking U.K.

Another alternative, as the Irish journalist Peter Geoghegan suggested, is that the island of Ireland could continue to function without a border, but anyone traveling from Northern Ireland to the rest of the U.K. would have to pass through immigration and customs checks. Such an arrangement would be short of formal unification but help prevent backsliding in the peace process.

On Friday evening, the UK government’s Northern Ireland Office released a statement making it clear that the secretary of state, Theresa Villiers — a Leave campaigner — had rejected the request for a referendum on Irish unity, arguing, “there is no reason to change our view that a majority of the people of Northern Ireland support the current political settlement and want to remain part of the UK.”