3. There exists an integer $n$ such that $\sqrt{p+n}+\sqrt{n}$ is an integer for the prime number $p$
(A) does not exist;
(B) there is only one;
(C) there are more than one, but a finite number;
(D) there are infinitely many.
3. (D)

Let $p$ be any odd prime, and $p=2k+1$. Then $n=k^{2}$, so we have
$$
\sqrt{p+n}+\sqrt{n}=\sqrt{2k+1+k^{2}}+\sqrt{k^{2}}=2k+1.
$$

Therefore, every odd prime has the property mentioned in the question, hence the answer is (D).