Donald Trump

Donald Trump was an American politician that the youth of 2016 considered to be unstoppable. (PROSE: What She Does Next Will Astound You)

Dara Morgan, a millionaire businessman from Derry, frequently wined and dined with "the Trumps, Gateses, de Rothschilds, Gettys and half a dozen more movers and shakers". (PROSE: Beautiful Chaos)

In December 2016, after discovering that the Jezrafeq were controlling the minds of most of humanity, Andy Hansen jokingly asked Erimem, "You don't think this could explain Brexit and Donald Trump, do you?". Later, while tracing the source of the mind control signal, Andy suggested that the signal may have been coming from Trump Tower "because [Donald Trump] and Nigel Farage want[ed] to piss [Andy] off even more before the year end[ed]". (PROSE: Cliff Richard Saves the World)

Behind the scenes
In DWM 503, Steven Moffat answered the question, "What does the Doctor make of Donald Trump's decision to run for President of the United States?" Moffat replied that "although the Doctor normally takes no part in politics, he is disgusted and turbulent with rage" and went on to say that the Doctor has seen a parallel universe in which Trump won the election and the world was plunged into "certain doom".

In the real world, Trump did in fact win the election. However, in the Shadow World 2017, made by the Monks to resemble the real world almost perfectly, an unnamed president was a dark haired individual who did not have much physical resemblance to Trump. According to Moffat, this episode was written before but filmed after the 2016 election.

Trump is implied, though not explicitly stated in dialogue, to be the current sitting US President in the following episode, The Pyramid at the End of the World. Bill notes that the recently elected President is "orange", and says that she wouldn't have voted for him. The final episode of the trilogy, The Lie of the Land, features an image of Trump clearly visible on one of the screens inside the room that houses their propaganda machine.