The paper aims to propose the fixed point property(FPP for short) of smallest open neighborhoods of the n-dimensional Khalimsky space and further, the FPP of a Khalimsky (K-, for short) retract. Let $(X, \kappa ^n_X)$ be an n-dimensional Khalimsky topological space induced by the n-dimensional Khalimsky space denoted by $(\mathbb{Z}^n, \kappa^n)$. Although not every connected Khalimsky topological space $(X, \kappa^n_X)$ has the FPP, we prove that for every point $x \in Z^n$ the smallest open K-topological neighborhood of x, denoted by $SN_K (x) \subset (\mathbb{Z}^n , \kappa^n)$, has the FPP. Besides, the present paper also studies the almost fixed point property (AFPP, for brevity) of a K-topological space. In this paper all spaces $(X, \kappa^n_X) := X$ are assumed to be connected and $| X | \ge 2$.