The growth properties of de Branges spaces and their subspaces are studied. It is shown that, for each given pair of growth functions λ(r) = O(r) and λ₁ = o(λ), there exist de Branges spaces of growth λ that have a de Branges subspace of growth λ₁. This phenomenon cannot occur for a class of de Branges spaces that, in a certain sense, behave regularly along the real axis.