Large language model assistants (LLM-assistants) present new opportunities to transform software development. Developers are increasingly adopting these tools across tasks, including coding, testing, debugging, documentation, and design. Yet, despite growing interest, there is no synthesis of how LLM-assistants affect software developer productivity. In this paper, we present a systematic review and mapping of 39 peer-reviewed studies published between January 2014 and December 2024 that examine this impact. Our analysis reveals that the majority of studies report considerable benefits from LLM-assistants, though a notable subset identifies critical risks. Commonly reported gains include accelerated development, minimized code search, and the automation of trivial and repetitive tasks. However, studies also highlight concerns around cognitive offloading and reduced team collaboration. Our study reveals that whether LLM-based assistants improve or degrade code quality remains unresolved, as existing studies report contradictory outcomes contingent on context and evaluation criteria. While the majority of studies (90%) adopt a multi-dimensional perspective by examining at least two SPACE dimensions, reflecting increased awareness of the complexity of developer productivity, only 15% extend beyond three dimensions, indicating substantial room for more integrated evaluations. Satisfaction, Performance, and Efficiency are the most frequently investigated dimensions, whereas Communication and Activity remain underexplored. Most studies are exploratory (59%) and methodologically diverse, but lack longitudinal and team-based evaluations. This review surfaces key research gaps and provides recommendations for future research and practice. All artifacts associated with this study are publicly available at https://zenodo.org/records/18489222