Named entity recognition is an important application within Danish NLP, essential within both industry and research. However, Danish NER is inhibited by a lack coverage across domains and entity types. As a consequence, no current models are capable of fine-grained named entity recognition, nor have they been evaluated for potential generalizability issues across datasets and domains. To alleviate these limitations, this paper introduces: 1) DANSK: a named entity dataset providing for high-granularity tagging as well as within-domain evaluation of models across a diverse set of domains; 2) and three generalizable models with fine-grained annotation available in DaCy 2.6.0; and 3) an evaluation of current state-of-the-art models’ ability to generalize across domains. The evaluation of existing and new models revealed notable performance discrepancies across domains, which should be addressed within the field. Shortcomings of the annotation quality of the dataset and its impact on model training and evaluation are also discussed. Despite these limitations, we advocate for the use of the new dataset DANSK alongside further work on generalizability within Danish NER.