Geospatial foundation models (GFMs) are reshaping remote sensing by shifting the field from task-specific modeling toward general-purpose geospatial intelligence. Mapping the damage to buildings stands to benefit substantially from this shift, as GFMs pretrained on large-scale Earth observation data provide generalizable geospatial representations that can facilitate effective responses to sudden-onset disasters. Although parameter-efficient fine-tuning (PEFT) facilitates GFM adaptation, parameter efficiency does not necessarily translate into label efficiency. Building damage mapping remains challenging due to scarce pre-existing labeled data or limited time for annotation. When weak supervision is introduced to alleviate the labeling bottleneck, we identify the difficulty of learning sufficiently discriminative representations as a key challenge for label-efficient GFM adaptation. To address this challenge, we propose a Positive-Unlabeled GFM Adaptation (PUMA) framework that requires annotations for only a small number of damaged buildings. More specifically, PUMA integrates non-contrastive weakly supervised representation refinement to mitigate the impact of noisy supervision on representation learning from a gradient perspective, and latent-state-based label disambiguation to enable reversible supervision refinement without requiring class prior estimation. Extensive experiments on disaster benchmark datasets from around the world demonstrated the effectiveness and robustness of PUMA. A real-world application to the 2025 Los Angeles wildfires further highlighted its potential to accurately map damage under highly limited supervision. More broadly, these findings suggest that weakly supervised PEFT offers a label-efficient pathway from GFMs to real-world applications.

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