Is the poly(A) tail added while transcription is still underway?
It's slightly more complicated than the comments above indicate.


  Glover-Cutter, K et al. (2007) RNA polymerase II pauses and associates with pre-mRNA processing factors at both ends of genes. Nature Structural &amp; Molecular Biology 15: 71-78


The authors present evidence that RNA polII pauses 0.5 - 1.5 kb downstream of the poly(A) site where processing factors are recruited (cleavage stimulation factor, or CstF, and cleavage polyadenylation specificity factor, CPSF). So the polymerase is still engaged with the template and the premRNA when processing is initiated, although it may not be actively elongating the transcript. 


  "Localisation of the active polII-3' end processing complex well downstream of the poly(A) consensus sequence agrees well with the cleavage of Chironomus BR1 transcripts after 600 bases of downstream sequence has been transcribed. This model is also supported by the fact that efficient poly(A) site cleavage requires an intact RNA tether linking the poly(A) site with the downstream polymerase." (references are cited in the original).


It is a semantic question as to whether you call this cotranscriptional processing.

[Incidentally this is yet another example which contradicts the commonly-stated idea that evolution acts to optimise everything to conserve ATP - here we have the transcription process discarding the energy from approximately 1000 phosphodiester bonds per transcript.]

UPDATE
@WYSIWYG The polymerase goes past the site where the poly(A) will eventually be added, and makes another 1000 bases of RNA before pausing to recruit factors that are needed for the subsequent cleavage at the pol(A) site which now lies "behind it". After that the poly (A) tail is added at the newly created 3' end. So the initiation of poly(A) addition (in the sense that the cleavage is an obligatory first step in the process) is accomplished while the polymerase is still engaged. I agree that the actual addition of the poly(A) cannot in any way be considered to be cotranscriptional. 

I was trying to make the point out that it isn't simply a case of the polymerase encountering the cleavage site and dropping off to allow a second unlinked process to take place.