  We explore the construction of four-dimensional thick branes supported by
massless 3-forms in a five-dimensional bulk space. The required residual
Poincar\'e symmetry on the brane is realised as a combination of the bulk
symmetries and the internal gauge symmetry of the 3-form. We show the presence
of a gradient instability for the perturbations orthogonal to the brane when
its profile decays slowly in the asymptotic regions. In particular, we
ascertain that the prevailing profiles found in the literature are susceptible
to such instabilities. We confirm our results by transitioning to the dual
formulation in terms of a shift-symmetric scalar. In this formulation, the
residual Poincar\'e group is trivially realised without internal symmetries,
and there is a diagonal translational invariance in the fifth dimension
reminiscent of the invariance under translations orthogonal to the brane of the
dual 3-form. We demonstrate the extension of our results to the cases of de
Sitter and Anti de Sitter branes where the gradient instabilities of
asymptotically slowly decaying branes persist. Finally, we briefly comment on
the construction of thick branes with massive 3-forms and their 1-form duals.

General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory