For constant $p>0$ there is no way to exactly sort $A$ with high probability. Think, e.g., of the first two elements: they will seem to be in the wrong order with probability $p$ and no other comparison provides any information on their order.
In fact, if an algorithm returns a permutation of $A$ that has maximum dislocation1 $D$ with high probability, then $D = \Omega(\log |A|)$.
The algorithm in the paper by Braverman and Mossel suggested by Yuval returns a permutation with maximum dislocation $O(\log |A|)$ and, in particular, returns the maximum-likelihood permutation. However, its running time can be quite large.
The following papers have improved the time complexity (in chronological order) for the problem of sorting with (asymptotically) optimal $O(\log |A|)$ maximum dislocation. Here the returned permutation does not need to be the maximum-likelihood one and $p$ needs to be smaller than some constant ($1/32$, $1/20$, or $1/16$):
1 The dislocation of an element in a permutation is the absolute difference between its position and its rank (position in the sorted sequence).