We study global injectivity of proper branched coverings from the open Euclidean 𝑛 -ball onto an open subset of the Euclidean 𝑛 -space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when 𝑛=3 or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen.