"Application-platform co-design" refers to the phenomenon of new platforms being created in response to changing application needs, followed by application design and development changing due to the emergence (and the specifics, limitations) of the new platforms, therefore creating, again, new application and platform requirements. This continuous process of application and platform (re-)design describes an engineering and management responsibility to constantly evaluate any given platform for application fit and platform-specific application design, and to consider a new or evolutionary platform development project due to evolving and changing application needs. In this paper, we study this phenomenon in the context of serverless computing and (big) data processing needs, and thus, for application-platform co-design for serverless data processing (SDP). We present an analysis of the state-of-the-art of function-as-a-service (FaaS) platforms, which reveals several configuration, deployment, execution, and measurement differences between popular platforms happening at-speed. These differences indicate already ongoing platform (re-)design processes resulting in more specialized serverless platforms and new, platform-specific challenges for application design. We discuss data processing needs of applications using the serverless model and present common initial (and undesirable) workaround solutions on the application level, giving additional argument to the creation of new SDP platforms. We present critical SDP requirements and possible new platform augmentations, but identify the need for engineering methods and tooling to better guide application-platform co-design. We argue to pay appropriate attention to the phenomenon of continuous application-platform co-design to better anticipate and to control future platform and application developments.