  In this work, we investigate the presence of thick branes modeled by a single
scalar field with Born-Infeld-like dynamics. We consider the 4-dimensional
metric being Minkowski, de Sitter or anti-de Sitter. We obtain the field
equations and the conditions to get a first order formalism compatible with
them. To illustrate our procedure, some specific models are presented. They
support localized warp factor and have their properties controlled by the
4-dimensional cosmological constant. In particular, a hybrid brane may arise,
with a thick or thin profile depending on the extra dimension being inside or
outside a compact space.
