I was aware that mature red blood cells (RBCs) lose their nucleus, ER and Golgi apparatus. However, I wasn't sure about the centrosome! Centrosomes, being MTOCs (microtubule organising centers), have known functions beyond those in cell division. Some are cell type specific! So be wary of thinking that all cells that do not undergo mitosis lack centrosomes. I find it flawed to reason along the lines of
mature RBCs are anucleated cells = no nucleus = no nuclear membrane = no mitosis = no mitotic spindle = no centrosomes
After a little research, I've found the following unreferenced statement in this opinion piece by Rieder, Faruki & Khodjakov from 2001 in Trends in Cell Biology:
In some cells such as megakaryocytes, myocytes and red blood cells of
higher animals, the centrosome degenerates. As a rule, however, such
cells are terminally differentiated and no longer capable of entering
the mitotic cycle.
If somebody finds a concrete example (or better yet, image!), that would be more convincing than an opinion piece.