Machine learning classification models are vulnerable to adversarial examples -- effective input-specific perturbations that can manipulate the model's output. Universal Adversarial Perturbations (UAPs), which identify noisy patterns that generalize across the input space, allow the attacker to greatly scale up the generation of these adversarial examples. Although UAPs have been explored in application domains beyond computer vision, little is known about their properties and implications in the specific context of realizable attacks, such as malware, where attackers must reason about satisfying challenging problem-space constraints. In this paper, we explore the challenges and strengths of UAPs in the context of malware classification. We generate sequences of problem-space transformations that induce UAPs in the corresponding feature-space embedding and evaluate their effectiveness across threat models that consider a varying degree of realistic attacker knowledge. Additionally, we propose adversarial training-based mitigations using knowledge derived from the problem-space transformations, and compare against alternative feature-space defenses. Our experiments limit the effectiveness of a white box Android evasion attack to ~20 % at the cost of 3 % TPR at 1 % FPR. We additionally show how our method can be adapted to more restrictive application domains such as Windows malware. We observe that while adversarial training in the feature space must deal with large and often unconstrained regions, UAPs in the problem space identify specific vulnerabilities that allow us to harden a classifier more effectively, shifting the challenges and associated cost of identifying new universal adversarial transformations back to the attacker.