Can different proteins be produced during translation of a single mRNA in eukaryotes?
There are multiple mechanisms that are known to lead to translation of substantially (or entirely) different proteins from a single mRNA. While these mechanisms are more typically seen in viruses, I'm focusing on examples documented within the endogenous transcriptome of eukaryotes. 

Alternative translation initiation

One process that can lead to different proteins being translated from the same mRNA in eukaryotes is the use of alternative translation initiation sites.1,2 Translation typically starts with a pre-initiation complex recognizing the 5' cap and loading onto the mRNA. This complex scans until it finds an appropriate start site. The choice of start site depends on the how well the ribonucleotide sequence matches the Kozak consensus sequence.3 If the region around the first AUG is not a good match for that consensus then a process known as leaky scanning occurs and the pre-initiation complex can continue along the mRNA until a "good" start site is found.

While this can result in proteins with different functions, they will typically still have large regions of amino acid sequence in common. One example of this is in a kinase known as MK2, which is "a key regulator of transcription, migration, death signaling and post-transcriptional gene regulation".4

Polycistronic mRNAs

While more common in prokaryotes, in some cases eukaryotes also have polycistronic transcripts5,6.  These transcripts encode multiple separate proteins (i.e. from independent open reading frames). The examples that I have found for mammals are all bicistronic (operons with two genes): LASS1-GDF1, SNRPN-SNURF, MTPN-LUZP6 and MFRP-C1QTNF5. You can search for those gene pairs, but there doesn't seem to be a huge amount of information available and in many cases one of the genes is almost completely uncharacterized. Eukaryotic operons (aka polycistronic mRNAs) are ubiquitous in trypanosomes, appear to be very common in nematodes (round worms), and are also frequently seen in Drosophila (a fly).6

Translational read-through

Translational readthrough aka. stop codon suppression occurs when an in frame stop codon is ignored either stochastically or under specific conditions and translation continues beyond that point. This leads to a C-terminal extension of the protein which in some cases has been shown to have functional implications7,8. An example of this is the [PSI+] prion in yeast, which promotes translational readthrough throughout the yeast transcriptome by inactivating a factor involved in translation termination.7 

Translational frameshifts

This is well covered in the answer by @Dirigible.

References:
1: Kochetov, A. V. (2008). Alternative translation start sites and hidden coding potential of eukaryotic mRNAs. Bioessays, 30(7), 683-691.
2: Wan, J., &amp; Qian, S. B. (2013). TISdb: a database for alternative translation initiation in mammalian cells. Nucleic acids research, 42(D1), D845-D850.
3: Acevedo, J. M., Hoermann, B., Schlimbach, T., &amp; Teleman, A. A. (2018). Changes in global translation elongation or initiation rates shape the proteome via the Kozak sequence. Scientific reports, 8(1), 4018.
4: Trulley, P., Snieckute, G., Bekker-Jensen, D., Menon, M. B., Freund, R., Kotlyarov, A., ... &amp; Gaestel, M. (2019). Alternative Translation Initiation Generates a Functionally Distinct Isoform of the Stress-Activated Protein Kinase MK2. Cell reports, 27(10), 2859-2870.
5: Tautz, D. (2008). Polycistronic peptide coding genes in eukaryotes—how widespread are they?. Briefings in Functional Genomics and Proteomics, 8(1), 68-74.
6: Blumenthal, T. (2004). Operons in eukaryotes. Briefings in Functional Genomics, 3(3), 199-211.
7: Schueren, F., &amp; Thoms, S. (2016). Functional translational readthrough: a systems biology perspective. PLoS genetics, 12(8), e1006196.
8: Loughran, G., Jungreis, I., Tzani, I., Power, M., Dmitriev, R. I., Ivanov, I. P., ... &amp; Atkins, J. F. (2018). Stop codon readthrough generates a C-terminally extended variant of the human vitamin D receptor with reduced calcitriol response. Journal of Biological Chemistry, 293(12), 4434-4444.