Solution:

The probability that a randomly chosen point has both coordinates less than $p$ is $p^{2}$, so the probability that at least one of its coordinates is greater than $p$ is $1 - p^{2}$. Since $p$ is the answer to this question, we have $1 - p^{2} = p$, and the only solution of $p$ in the interval $[0,1]$ is $\frac{\sqrt{5} - 1}{2}$.