We present a novel method, DistillAdapt, for the challenging problem of Source-Free Active Domain Adaptation (SF-ADA). The problem requires adapting a pretrained source domain network to a target domain, within a provided budget for acquiring labels in the target domain, while assuming that the source data is not available for adaptation due to privacy concerns or otherwise. DistillAdapt is one of the first approaches for SF-ADA, and holistically addresses the challenges of SF-ADA via a novel Guided Attention Transfer Network (GATN) and an active learning heuristic, H_AL. The GATN enables selective distillation of features from the pre-trained network to the target network using a small subset of annotated target samples mined by H_AL. H_AL acquires samples at batch-level and balances transfer-ability from the pre-trained network and uncertainty of the target network. DistillAdapt is task-agnostic, and can be applied across visual tasks such as classification, segmentation and detection. Moreover, DistillAdapt can handle shifts in output label space. We conduct experiments and extensive ablation studies across 3 visual tasks, viz. digits classification (MNIST, SVHN), synthetic (GTA5) to real (CityScapes) image segmentation, and document layout detection (PubLayNet to DSSE). We show that our source-free approach, DistillAdapt, results in an improvement of 0.5% - 31.3% (across datasets and tasks) over prior adaptation methods that assume access to large amounts of annotated source data for adaptation.