Is there an example of an oracle A such that P = NP but $\mathsf{P}^A\neq\mathsf{NP}^A$?
Jonathan Katz explains the original construction due to Baker, Gill and Solovay in his lecture notes.

The idea is to construct such a language by diagonalization. The first step is to show the lemma that for any deterministic, polynomial-time oracle machine $M$, you can construct a language $B$ so that $M^B$ (that is, $M$ with oracle access to $B$) does not decide the language $L_B = \{1^n\textrm{ | }B \textrm{ contains a string of length } n\}$. For sufficiently large $n$, $M$ can not query all strings of length $n$ of $B$, so it is possible to "hide" (or not) a string of length $n$ in $B$ so that $M$ gives the wrong answer.

Note that $L_B$ is in $NP$ since for a given $n$ we can guess the string of length $n$ (if any) that is in $B$ and then query the oracle once.

This doesn't give the desired result (it says that for any deterministic, polynomial-time TM there is a language rather than for any language there is a TM). However, we can enumerate all deterministic, polynomial Turing oracle machines $M_1,M_2,M_3,\ldots$. We can (by the lemma) construct a language $B_1$ for which $M_1^{B_1}$ fails to decide $B_1$. Using a construction similar to in the lemma, we can then modify $B_1$ to obtain $B_2$ so that both $M_1^{B_2}$ and $M_2^{B_2}$ fail to decide $B_2$, and so on... 

We eventually obtain a language $B_{\infty}$ so that $L_{B_{\infty}}$ is in $NP^{B_{\infty}}$, but not in $P^{B_{\infty}}$