are there known cases where $\binom{n}{k}$ is a perfect prime power?
Solution 1:
Erdos proved (in 1951) the stronger result that $\binom{n}{k}$ is never a perfect power if $k > 3$ (assuming that $n \geq 2k$ as we may). His argument is completely elementary -- see https://www.renyi.hu/~p_erdos/1951-05.pdf.
For $k=2$ and $k=3$, the case of possible prime powers is easily treated by divisibility arguments. The more general situation of arbitrary perfect powers for these values of $k$ was handled by Gyory in 1997 (in Acta Arithmetica), by a rather more complicated proof.