Column type annotation is the task of annotating the columns of a relational table with the semantic type of the values contained in each column. Column type annotation is a crucial pre-processing step for data search and integration in the context of data lakes. State-of-the-art column type annotation methods either rely on matching table columns to properties of a knowledge graph or fine-tune pre-trained language models such as BERT for the column type annotation task. In this work, we take a different approach and explore using ChatGPT for column type annotation. We evaluate different prompt designs in zero- and few-shot settings and experiment with providing task definitions and detailed instructions to the model. We further implement a two-step table annotation pipeline which first determines the class of the entities described in the table and depending on this class asks ChatGPT to annotate columns using only the relevant subset of the overall vocabulary. Using instructions as well as the two-step pipeline, ChatGPT reaches F1 scores of over 85% in zero- and one-shot setups. To reach a similar F1 score a RoBERTa model needs to be fine-tuned with 300 examples. This comparison shows that ChatGPT is able deliver competitive results for the column type annotation task given no or only a minimal amount of task-specific demonstrations.