However, the ballot's effect on Swiss government decisions will be limited by the country's political system, which requires broad consensus among politicians and frequently subjects policy decisions to national referendums. The leftwing Social Democratic party held its vote steady at 19 per cent, with more centrist parties losing support.
The SVP is one of Europe's longest established, ultraconservative political parties. As well as arguing for tougher controls on immigration, it opposes Switzerland joining the EU. Christoph Blocher, a wealthy businessman and its veteran leader, sprung to international attention in the early 1990s when he prevented EU membership.
During the election campaign, the SVP warned of "asylum chaos" across Europe, and Mr Blocher told the Financial Times that the refugee crisis had shown the Schengen agreement on free movement of people across Europe — of which Switzerland is a signatory — to be unworkable
The SVP has also complained that Swiss authorities have given in too quickly to international pressure, for example on bank secrecy laws. But its ability to reshape policies will depend on whether it can win a second seat in the seven-person "federal council", which acts like a cabinet government and includes representatives from across the political spectrum.
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The SVP has had only one seat on the council since it threw Eveline Widmer-Schlumpf, finance minister, out of the party after she agreed to take office at the behest of the other main political parties.
Sunday's results showed Ms Widmer-Schumpf's breakaway Conservative Democratic party losing support — jeopardising her chances of remaining in office. "We will still have a consensus government, but the election [on December 9] of the federal council could be thrilling," said Daniel Kalt, UBS's chief economist in Switzerland.
Last year, the SVP last year narrowly won a referendum demanding quotas on immigration from EU as well as non-EU countries. Implementing such a move had threatened a clash with Brussels, because it would breach the principle of free movement of people. It has made more complicated efforts to renegotiate Switzerland's relationship with the EU, which is based on a web of bilateral agreements.
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Electoral support for Switzerland's political right was also strengthened by worries about the effect of the steep appreciation of the franc after the Swiss central bank in January abandoned attempts to cap its value against the euro. The threat to the export-led economy initially boosted support for the business-orientated Free Democratic party. But as Europe's refugee crisis intensified over the summer, support for the SVP also rose.