$\newcommand{\R}{\mathbb R}\newcommand\ep\epsilon\newcommand\tsi{\tilde\sigma}$Yes, of course. This follows by the multivariate (here, bivariate) so-called delta method. 

Indeed, we may assume that 
\begin{equation*}
	X=\sum_{i=1}^n X_i,\quad Y=\sum_{i=1}^n Y_i,
\end{equation*}
where $X_1,Y_1,\dots,X_n,Y_n$ are independent identically distributed (iid) Bernoulli random variables (r.v.'s) with parameter $p\in(0,1)$. 

For each $i\in[n]:=\{1,\dots,n\}$, let 
\begin{equation*}
	V_i:=(X_i-p,Y_i-p),
\end{equation*}
so that $V_1,\dots,V_n$ are iid zero-mean random vectors in $\R^2$. Then 
\begin{equation*}
	XY=n^2 f(\bar V)+n^2p^2, \tag{1}\label{1}
\end{equation*}
where $\bar V:=\frac1n\,\sum_{i=1}^n V_i$ and for $(x,y)\in\R^2$ 
\begin{equation*}
	f((x,y)):=f(x,y):=(x+p)(y+p)-p^2,
\end{equation*}
so that $f(0,0)=0$ and for $L:=f'(0,0)$ we have 
\begin{equation*}
	L(x,y)=px+py
\end{equation*}
and 
\begin{equation*}
	|f(x,y)-L(x,y)|=|xy|\le\tfrac12\,\|(x,y)\|^2,
\end{equation*}
where $\|(x,y)\|:=\sqrt{x^2+y^2}$. 
So, condition (2.1) of [this paper][1] holds (for any real $\ep>0$ and $M_\ep=1$). 

Moreover, $v_3:=E\|V_1\|^3<\infty$ and 
\begin{equation*}
	\tsi:=\sqrt{EL(V_1)^2}=\sqrt{p^2 E(X_1-p+Y_1-p)^2}=\sqrt{2p^3q}>0,
\end{equation*}
where $q:=1-p$. 
So, by \eqref{1} and Theorem 2.9 of the [same paper][1], 
\begin{equation*}
	\frac{XY-n^2p^2}{n^2\sqrt{2p^3q/n}}
	=\frac{f(\bar V)}{\tsi/\sqrt n}\to Z\sim N(0,1) \tag{2}\label{2}
\end{equation*}
(in distribution as $n\to\infty$). 

Note also that $EXY=n^2p^2$ and 
$$Var\,XY=2n^3p^3q+n^2p^2q^2=2n^3p^3q(1+O(1/n))\\ 
\sim2n^3p^3q=(n^2\sqrt{2p^3q/n})^2.$$ 
Thus, by \eqref{2}, 
\begin{equation*}
	\frac{XY-EXY}{\sqrt{Var\,XY}}\to Z,   
\end{equation*}
as desired. 

Moreover, it follows from cited Theorem 2.9 that 
\begin{equation*}
	\sqrt n\,\sup_{z\in\R}\Big|P\Big(\frac{XY-EXY}{\sqrt{Var\,XY}}\le z\Big)-P(Z\le z)\Big|\le C_p
\end{equation*}
for some real $C_p>0$ depending only on $p\in(0,1)$ and all natural $n$. 
 

[1]: https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-10/issue-1/Optimal-order-bounds-on-the-rate-of-convergence-to-normality/10.1214/16-EJS1133.full