That is currently not possible and actually a limitation of the Elastic Load Balancing (ELB) in use, see e.g. Why does Amazon not allow assigning an Elastic IP to an Elastic Load Balancer?:
The elastic load balancer is simply a collection of individual
software load balancers hosted within EC2, with DNS load balancing
traffic among them. The pool can contain many IPs, at least one per
availability zone, and depending on traffic levels.
D.Svanlunds's answer to Load balancers and elastic IP provides a good summary of why ELB is designed this way, and Shlomo Swidler's The “Elastic” in “Elastic Load Balancing”: ELB Elasticity and How to Test it provides an early but in-depth analysis of the ELB architecture and the implications for testing a web application in this context.
Regarding your comment: You are communicating with your AWS Elastic Beanstalk application via the DNS name mapped to the Elastic Load Balancer, to which you can map a CNAME record in turn as usual, see Overview of Elastic Load Balancing:
Elastic Load Balancing automatically generates a DNS name for each
load balancer. You can map any other domain name (such as
www.example.com) to the automatically generated DNS name using CNAME
or use an Amazon Route 53 alias for the load balancer's DNS name.
Further details can be found in Using Domain Names With Elastic Load Balancing, which describes how to associate your Elastic Load Balancing instance with a custom domain name — including subdomain names and the zone apex.