Space Information Networks (SIN) allow terrestrial mobile nodes to gain access to the Internet anywhere and at any time. Due to its dynamic network coverage, it has been gaining wider attention from researchers and network service providers. In comparison to terrestrial networks, the majority of data transmission in SIN is through air interface, which makes it vulnerable to adversarial attacks such as illegal data access, eavesdropping attack, etc. To prevent unauthorized user access, this research work focuses on designing a secure and lightweight access authentication mechanism using extended Chebyshev chaotic maps for resource-constrained mobile nodes as an act of the first line of defense. The proposed scheme achieves several security functionalities, including resistance to impersonation attack, session key secrecy, mutual authentication, and key leakage attack. Furthermore, the soundness of the proposed scheme is validated using the ProVerif tool. Finally, the performance analysis depicts that the proposed scheme is highly efficient compared to existing methods.