zoology - Is the appendix a vestigial structure in all vertebrates?

Aside from humans, it is largely rodents and most notably rabbits that have an appendix. Therefore, using rabbits as my example:

In rabbits, the appendix is thought to have a key role in the development of the immune system. Specifically it has been shown experimentally that when neonatal rabbits are given an appendectomy levels of Immunoglobulin A and G (IgA/IgG) fall dramatically. Both these polypeptides are prominent antibodies - IgA plays a key role in mucosal immunity whilst IgGs in humoral immunity. These effects were localised to the small intestine of the rabbits, however were statistically significant. ¹

In infants, the rabbit appendix resembles the chicken bursa and sheep ileal Peyer's patch (both performing similar functions as described above).²

This has led to some ongoing research as to whether the human appendix has a similar properties in having function in infants ³:

If the human appendix functions as a primary lymphoid organ, it may occur during the first few months of age when the GC T-cell density is low.

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¹Neonatal appendectomy impairs mucosal immunity in rabbits. Cell Immunol. 1997 Nov 25;182(1):29-37

²The appendix functions as a mammalian bursal equivalent in the developing rabbit. Adv Exp Med Biol. 1994;355:249-53.

³A morphological and immunohistological study of the human and rabbit appendix for comparison with the avian bursa. Dev Comp Immunol. 2000 Dec;24(8):797-814.