While I can't prove for certain that it beats the current top two contenders (USS Stargazer with 10, and IKS Rotarran with 9 or 13), Chakotay's Maquis ship, the Val Jean, is another strong contender.
Oddly enough, it's name is never spoken (making transcript searches much more difficult!), but it is seen on a computer screen in one episode. While its crew is referenced heavily throughout the series, I count at least 10 or 11 episodes where the actual ship is mentioned (depending on if you count Caretaker as one or two episodes). Without rewatching the whole series, I can't guarantee there aren't more.
1x01/02 - Caretaker - The ship has a prominent role, and is destroyed.
1x03 - Parallax - Chakotay asks Janeway if she'd have served him if they "were on the Maquis ship now instead of Voyager"
1x11 - State of Flux - Chakotay, commenting on Tuvok's and Seska's both being spies, asks "Was anyone on board that ship working for me?"
1x16 - Learning Curve - An ex-Maqui tells Tuvok "On our ship, when something went wrong, you fixed it," to which Tuvok replies he is "no longer on a Maquis ship."
2x05 - Non Sequiter - Paris tells Kim that Janeway asked him to "help her track down that Maquis ship".
5x24 - Relativity - An admiral assures Janeway that Tuvok is "aboard Chakotay's ship."
6x09 - The Voyager Conspiracy - Seven believes Chakotay was deliberately trying to use the Caretaker array as a weapon against the Federation, "offering his vessel as bait". Janeway tells Seven that "Tuvok was a spy on Chakotay's ship."
6x25 - The Haunting of Deck Twelve - Chakotay admits he "used to have long conversations with my Maquis ship."
7x04 - Repression - The name of the ship, Val Jean, is seen on a computer screen.
7x11 - Shattered - Janeway tells a time-displaced Chakotay that "we're about to chase your ship into the Badlands".
Of course, admitting oblique references like "my/your/that ship" might also increase the number for some of the other contenders.
EDIT: Sticking with the same series, and going for runner-ups, Neelix's "freighter" the Baxial, was apparently shown or mentioned in 8 different episodes, according to memory alpha. Granted, this may count as a support ship or shuttle, given that it fit within Voyager's shuttlebay.