We prove the existence and attraction property of an unstable manifold for solutions to the partial neutral functional differential equation of the form $$\left\{\begin{array}{ll} \frac{\partial}{\partial t}Fu_{t}= B(t)Fu_{t} +\varPhi(t,u_{t}),\quad t\ge s;~ t,s\in \mathbb{R},\\ u_{s}=\phi\in \mathcal{C}:=C([-r, 0], X) \end{array}\right.$$ under the conditions that the family of linear operators $(B(t))_{t\in \mathbb{R}}$ defined on a Banach space $X$ generates the evolution family $(U(t, s))_{t\geq s}$ having an exponential dichotomy on the whole line $\mathbb R$, the difference operator $F:\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\| \Phi (t,\phi ) -\Phi (t,\psi )\| \le \phi (t)\|\phi -\psi \|_{\mathcal{C}}$ for $\phi ,~ \psi \in \mathcal{C}$, where $\varphi(\cdot)$ belongs to an admissible function space defined on $\mathbb R$. Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing $F$-induced trajectories.