Prelude: Thinking about protein synthesis
I remember the time when there were only two sites on the ribosome, and when it became clear there were more, I resented the need to make my lecture notes and my contribution to a venerable text book more complicated. So I am neither a devotee nor an expert on the exit site, but if asked to guess who was right (even if, or especially if, the student was supported by Campbell’s Biology as @CSharp informs us) I would put my money on the ‘professor’. Why? Because protein synthesis and the ribosome are so ancient that although there have been changes in eukaryotic proteins synthesis, something so fundamental as the sites for the fundamental process would not change.
Answer
Yes, Virginia, the eukaryotic ribosome does have an exit site.
The evidence
Although editions of the venerable text book no longer appear and my research no longer impinges on protein biosynthesis, I still collect papers on the subject with my bibliographic software. The latest relevant publication I have is by Khatter et al. from Nature in 2015, entitled Structure of the Human 80S Ribosome.
As not all list members have access to Nature, as a subscriber I have taken the liberty to reproduce a a section from the abstract (my emphasis) and a pertinent illustration:
Here we report the near-atomic structure of the human ribosome derived
from high-resolution single-particle cryo-electron microscopy and
atomic model building. The structure has an average resolution of 3.6A
̊, reaching 2.9A ̊ resolution in the most stable regions. It provides
unprecedented insights into ribosomal RNA entities and amino acid side
chains, notably of the transfer RNA binding sites and specific
molecular interactions with the exit site tRNA.

The cryo-EM map and atomic coordinates have been deposited in the EMDB and Protein Data Bank under accession codes EMD-2938 and 4ug0, respectively.
I admit I haven’t researched and rehearsed the evidence, but I think that would be another question.