  We show that our previous work on Galilei and Carroll gravity, apt for
particles, can be generalized to Galilei and Carroll gravity theories adapted
to p-branes (p = 0, 1, 2, ...). Within this wider brane perspective, we make
use of a formal map, given in the literature, between the corresponding p-brane
Carroll and Galilei algebras where the index describing the directions
longitudinal (transverse) to the Galilei brane is interchanged with the index
covering the directions transverse (longitudinal) to the Carroll brane with the
understanding that the time coordinate is always among the longitudinal
directions. This leads among other things in 3D to a map between Galilei
particles and Carroll strings and in 4D to a similar map between Galilei
strings and Carroll strings. We show that this formal map extends to the
corresponding Lie algebra expansion of the Poincar\'e algebra and, therefore,
to several extensions of the Carroll and Galilei algebras including central
extensions. We use this formal map to construct several new examples of Carroll
gravity actions. Furthermore, we discuss the symmetry between Carroll and
Galilei at the level of the p-brane sigma model action and apply this formal
symmetry to give several examples of 3D and 4D particles and strings in a
curved Carroll background.
