No diagnosis can be conclusive based solely on a public persona
Someone's public persona is not always indicative of their internal mental health.
Mental illnesses exist on a spectrum. Even if Trump displays some traits of a mental health disorder, it is not definitive proof he has it.
Modern research has indicated that mental illness is not a black and white issue. It is not a case of an individual either has a narcissistic personality disorder or they do not. These disorders exist on a spectrum.
It is undeniably true that Donald Trump has some of the traits that are present in those with narcissistic personality disorders. However, there is no definitive point at which we can say yes he has it or no he does not. It is not enough to simply say Donald Trump is showing the symptoms of a mental disorder. Because many people show these symptoms but do not have the disorder. Many mental health professionals assert that these traits only become a disorder when they have a detrimental effect on someone's life. If Trump's inflated sense of self-importance were to be damaging his relationships with his family and friends, it may be signs of mental illness. However, there has been no suggestion that his traits are damaging relationships. Nobody has asked his family members questions about their relationship with Donald Trump. Even if they did, the Trump family may not publicly reveal these details. Therefore, it is impossible to say if Donald Trump's narcissistic personality traits are indicative of a narcissistic personality disorder.
At a certain point, showing the traits of a mental health disorder becomes a diagnosis of having a mental health disorder, otherwise nobody would ever receive a diagnosis. If that threshold is that it is damaging to the relationships around you, then that threshold has undoubtedly been reached with Trump. His White House is full of people that have anonymously expressed concerns over his mental health. This is a sign of strained relationships with those around him. Additionally, his inflated sense of self-importance and emphasis on personal relationships over the interests of the American people (particularly in areas like foreign policy) are damaging his relationship with government institutions. He openly rejected the findings of his own national intelligence institutions on Russian interference in the 2016 US election because Vladimir Putin was "extremely strong and powerful in his denial". This damaged his relationship with the intelligence community.  He has fallen out with and publicly insulted plenty of people who were at one point close to him, including; Steve Bannon, Michael Cohen, Omarosa Manigault, George Papadopoulos, Barbara Res, Tony Schwartz and Rex Tillerson. Whatever the threshold is for mental illness, Donald Trump's behaviour and inability to maintain relationships clearly indicates that he meets it.
[P1] Mental illness occurs on a spectrum, not a binary scale. [P2] It is impossible to determine at what point Trump's personality becomes a mental illness. [P3] Therefore, it is impossible to know if Trump is mentally ill.
[Rejecting P2] It is possible to determine that point and Trump surpasses it.