Well there is the press, the Washington Post from what I'm told did a good job of unpacking his lies, which is why he accuses them of being 'fake news', 'fake media' and 'an enemy of the people'.
To legally 'punish' someone (including the President), one must try them first for some indictable crime and he/she must be convicted; now, according to Article 1.2 of the Constitution it is only the Senate that has the power to do this, and in fact clause 1.4 states:
The President, the Vice-President and all civil officers of the United States shall be removed from office on impeachment for, and conviction of, treason, bribery or other high crimes and misdemeanours
Note the term 'high crime'; this means that the trial is outside the ordinary judicial arrangements; and in fact formal charges would be brought by the judiciary committee of the House of Representatives, and the trial itself is held by the Senate itself - that is the jury is the Senate (the Supreme Court is only involved in upholding due process).
Now, you ask whether:
the President can be punished for lying to the people
You aren't (obviously) asking for lying in a general sense but 'lying to the people'; and nor are you (obviously, again) asking for Trump to be considered in his own person as a citizen, but as a holder of an office - the office of the Presidency, for you say POTUS and not Mr Trump.
Now what does it mean for the President to lie to the People? It's a phrase that one might find in a literary work, say a political drama or perhaps in an actual political speech; in that context it's usually understood as generally signalling a high crime.
So, in this sense and with this qualifier, then yes.
It's worth noting that President Nixon who resigned rather than be impeached had ordered an illegal wiretap on a political party and then lied about it (he was accused of a 'coverup'), here lying was part of the crime; also had it gone to the House Judiciary Committee, who would have drawn up the formal charge sheet (the Articles of Impeachment) it could well have been likely that one of the formal charges would be the 'coverup' itself.
Hence, in this particular sense too.