Constructing APN or 4-differentially uniform permutations achieving all the necessary criteria is an open problem, and the research on it progresses slowly. In ACISP 2011, Carlet put forth an idea for constructing differentially uniform permutations using extension fields, which was illustrated with a construction of a 4-differentially uniform (n,n)-permutation. The permutation has optimum algebraic degree and very good nonlinearity. However, it was proved to be a permutation only for n odd. In this note, we investigate further the construction of differentially uniform permutations using extension fields, and construct a 4-differentially uniform (n,n)-permutation for any n. These permutations also have optimum algebraic degree and very good nonlinearity. Moreover, we consider a more general type of construction, and illustrate it with an example of a 4-differentially uniform (n,n)-permutation with good cryptographic properties.