Several candidates have been tipped as potential successors to Adalbert Durrer, who resigned as president of the Christian Democratic Party on Thursday. A leading political analyst said the next leader would need to be "superman" to hold the party together.
Under Durrer's leadership, the party made a significant shift towards the right, exposing fault lines on issues such as the proposed legislation of abortion and the timetable for negotiations on Swiss membership of the European Union.
Durrer himself blamed internal party divisions for his surprise resignation after four years at its helm, and analysts agree that, unless a strong leader emerges to replace him, the splits are likely to widen, threatening the party's long-term future.
Adrian Vatter, a political scientist at the University of Bern, told swissinfo that his successor will face an almost impossible task to reunite the centre and the right.
"The next step will be to find a superman or superwoman who can lead this party. There are a lot of conflicts. We saw this in the recent debate over abortion, which showed how difficult it is for a party leader to bridge divides on these key issues."
One party member tipped to succeed Durrer is Lucrezia Meier-Schatz, a parliamentarian from canton St Gallen. She is keeping tight-lipped about her chances, but is clearly in favour of moving the party back towards the centre.
"Durrer tried to orient the party in a certain way, and his successor will have somehow to reunite it. He or she must be a person who is able to have a constructive dialogue with everyone in the party - someone from the centre."
The Christian Democrats share power with three other parties in government, among them the Social Democrats. Andreas Gross, a parliamentarian from Zurich, says the party won't survive unless the next leader broadens its appeal.
"I hope the party finds a president who is more open-minded, more serious, more modern, and who doesn't think the party can survive by holding on to its traditional base in central Switzerland."
But in the longer term, political analyst Adrian Vatter doubts the party has a future at all. He predicts that increasing numbers of disillusioned supporters will defect to other parties, and that eventually the two centre-right parties in government - the Christian Democrats and the Radicals - will give way to a new centrist grouping.
"I think we will eventually have only one big party - perhaps formed from a merger between the two centre parties."
Part of the reason, he says, has to do with a changing political landscape, which threatens the existence of all parties with religious origins.
"Thirty or 40 years ago, if you were Catholic, you joined the Christian Democrats. Now, especially in the eastern part of Switzerland, many Catholics vote for the [rightwing] Swiss People's Party. In a way, parties with religious origins have become obsolete."