As far as I remember, no.
The only death that happens in the cartoon is on the episode Blue Cat Blues, where it is implied that both of them commit suicide at the end of the episode (it is not actually shown).
From Controversial Cartoon Deaths:
This is not the typical episode about Tom & Jerry. Tom falls madly in
love with another cat and showers his sweetie with all sorts of gifts
including jewelry and even a car -- but she doesn't seem to be
impressed with his meagre fortune. He even signs away his arm and a
leg for her, but alas, to no avail. When things go awry, he later
becomes an alcoholic. Why? Mr. Butch (Tom's rival) stole his love away
The heartbroken Tom winds up in the gutter and sadly, Mr. Butch ends
up marrying Tom's only true love. Of course, all of this is narrated
by a 1950's deadpan voice, which adds to the eerie feeling of the
The episode ends off with Tom on the train tracks with a train
approaching... This is definitely not a pick-me-upper.