In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.