Nope - and this is a key point. The framers intended there to be a mechanism to hold a president accountable - impeachment. But it is badly formed, and has never been used, even though arguably it should have been in every case it was used in.

Also impeachment only removes them from office which is far from a complete punishment. They're still completely free people after impeachment, the only right that's taken away is the ability to be President which is a pretty paltry punishment especially for second term Presidents.

There have been impeachment proceedings started against four presidents, all of which would qualify as it being used, even if the presidents weren't successfully impeached. Impeachment proceedings have been completed against 3 presidents and in all cases the senate vote failed to meet the 2/3 supermajority. An impeachment inquiry was also started against Nixon, but he resigned before the house could vote. One could argue that the impeachment inquiry in that case was successful.

The procedure for impeaching federal judges is similar, in that the vote needs first to pass the house, then a supermajority of the senate. In 2009, federal judge Thomas Porteous Jr, was successfully impeached by congress. Congress then also voted to prohibit him from holding future federal office.

https://guides.loc.gov/federal-impeachment/thomas-porteous

The problem is that impeachment is a political process, not a legal or judicial process. I feel like using the term "prosecuting" to describe an impeachment trial in the Senate is a bit of a misnomer.

Regardless, impeachment is there to remove the president from power. It can't jail or otherwise punish the president. The regular old legal system should be doing that, for everyone, regardless of whether or not they're an elected official.