No, the CLT need not hold under these assumptions. Consider the following example: take $p=1/2$ for definiteness, and divide the (discrete) time into intervals $I_1=[1,2]$, $I_n=(2^{n-1}, 2^n]$, $n\geq 2$. On each interval, let $(X_k, k\in I_n)$ be the i.i.d. Bernoulli($1/2$) *conditioned* on $\sum_{k\in I_n}X_k=|I_n|/2$ (think about placing $|I_n|/2$ balls into $|I_n|$ urns at random); clearly, $(X_i,X_j)$ are negatively correlated when $i,j\in I_n$ for some $n$. Then the CLT doesn't hold since the sum becomes deterministic from time to time.

The correlations are not *strictly* negative though (because the $X$'s are independent if belong to different intervals), but you can probably make them so by some "small" perturbation.