Several hundred demonstrators marched in the capital Bern last Saturday to protest against President Trump's planned visit to the World Economic Forum (WEF) meeting in Davos.
One protestor carried a banner with the image of the US president and a red nuclear button that read ‘Kill Trump with his own weapons', the Swiss news agency SDA reported.
As a result, the police have now opened an investigation into public incitement to commit a crime, it said.
This comes under article 259 of the criminal code, and can be punished by up to three years imprisonment or a fine.
A spokesman for Bern cantonal police told SDA the police had checked with the prosecutor's office to see whether the incident could infringe article 259 before opening the investigation.
Last weekend's protest by left-wing demonstrators passed off peacefully, and police did not arrest the placard-carrier in order to avoid any escalation, SDA said.
In March last year the police opened a criminal investigation in a similar case when a Kurdish protestor against the Turkish President Recep Tayyip Erdogan carried a ‘Kill Erdogan' placard at a demonstration in Bern.
That investigation is still ongoing.
A number of demonstrations are taking place ahead of the WEF meeting to be held in Davos from January 23rd to 26th.
A group of Americans in Switzerland is staging a public event in Zurich on Saturday to apologize for the “damage that the current US administration is trying to cause around the world”.