A Wireless Body Area Network (WBAN), introduced into the healthcare sector to improve patient care and enhance the efficiency of medical services, also brings the risk of the leakage of patients’ privacy. Therefore, maintaining the communication security of patients’ data has never been more important. However, WBAN faces issues such as open medium channels, resource constraints, and lack of infrastructure, which makes the task of designing a secure and economical communication scheme suitable for WBAN particularly challenging. Signcryption has garnered attention as a solution suitable for resource-constrained devices, offering a combination of authentication and confidentiality with low computational demands. Although the advantages offered by existing certificateless signcryption schemes are notable, most of them only have proven security within the random oracle model (ROM), lack public ciphertext authenticity, and have high computational overheads. To overcome these issues, we propose a certificateless anonymous signcryption (CL-ASC) scheme suitable for WBAN, featuring anonymity of the signcrypter, public verifiability, and public ciphertext authenticity. We prove its security in the standard model, including indistinguishability, unforgeability, anonymity of the signcrypter, and identity identifiability, and demonstrate its superiority over relevant schemes in terms of security, computational overheads, and storage costs.