A psychiatrist responsible for the DSM definition says he does not, but the reasons given are not very good ones. (See an earlier discussion of this topic here.) He claims that Trump does not suffer distress and impairment connected to his personality traits (note that the psychiatrist does not deny he manifests those traits). He appears to confuse the fact that money and lawyers shield him from the worst consequences of his psychological peculiarities with the fact that he otherwise would be severely impaired. And the impairment is apparent in other ways: he doesn't appear to have real human relationships (he's on his third purchased wife), his relations with others are wholly instrumental, and despite inheriting a huge fortune and investing in New York real estate in the 1970s, he was never able to become a real player in the New York real estate market because others avoided him because of his problems. (This bears emphasizing: he was never one of the top ten developers in New York, and those folks would not deal with him. Anyone familiar with the NYC real estate market knows this!) We have less evidence about the distress he suffers, but it is hard not to interpret the bizarre tweeting and other outbursts as signs of serious distress, which he then tries to mend via public displays of various kinds.
Psychiatry is an epistemically feeble discipline, and the connotations of "mental illness" complicate things further. Let's drop the question of whether his narcissistic personality disorder is a mental illness. What's useful about the personality disorders, when they are useful at all, is that they do identify stable patterns of behavior that some people do exemplify. Trump is one. Having a narcissistic personality disorder may or may not be incompatible with being President--so far, the evidence is that he's quite impaired in his ability to govern, but that's all to the good given the mischief he would otherwise pursue were he more competent.