Is there any number with $2017$ divisors whose sum of digits is $2017$?
We know that $2017$ is prime and any number satisfying the required condition is of the form $p^{2016}$, where $p$ is a prime number. From here I couldn't make any progress.
Any help or reference would be appreciated.
Heuristic argument
You have correctly identified that such a number must be of the form $p^{2016}$, where $p$ is a prime.
From Wolfram, $$\begin{array}{c|c}\text{number}&2^{2016}&3^{2016}&5^{2016}&7^{2016}&11^{2016}&13^{2016}&17^{2016}&19^{2016}\\\hline\text{sum of digits}&2656&4293&6211&7552&9559&10126&11539&11584\end{array}$$ Hence we would not expect any number to have such properties; otherwise at least $2746-2017=729$ digits have to be $0$ for the next possible number $23^{2016}$ (and every other digit must be $1$), which is extremely unlikely.