Protein design holds immense potential for optimizing naturally occurring proteins, with broad applications in drug discovery, material design, and sustainability. How-ever, computational methods for protein engineering are confronted with significant challenges, such as an expansive design space, sparse functional regions, and a scarcity of available labels. These issues are further exacerbated in practice by the fact most real-life design scenarios necessitate the simultaneous optimization of multiple properties. In this work, we introduce ProteinNPT, a non-parametric trans-former variant tailored to protein sequences and particularly suited to label-scarce and multi-task learning settings. We first focus on the supervised fitness prediction setting and develop several cross-validation schemes which support robust perfor-mance assessment. We subsequently reimplement prior top-performing baselines, introduce several extensions of these baselines by integrating diverse branches of the protein engineering literature, and demonstrate that ProteinNPT consistently outperforms all of them across a diverse set of protein property prediction tasks. Finally, we demonstrate the value of our approach for iterative protein design across extensive in silico Bayesian optimization and conditional sampling experiments.