Are the ice skaters in Yuri On Ice based on real life counterparts? I watched the first few episodes of Yuri On Ice! and my girlfriend who iceskates said that she thought that the character Yuri Plisetsky really reminded her of Yulia Lipnitskaya in that they shared similar features and backgrounds. 

Are they connected? What about the other characters?
Yes, most skaters in Yuri!!! on Ice are based on real-life counterparts.
There are some official sources regarding the model of Yuri!!! on Ice's characters. According to this article,

Yuri Plisetsky and Yulia Lipnitskaya
Sharing the same name as the show’s protagonist, fifteen-year-old Yuri is a extremely talented and at times foul-mouthed Russian skater. He is making his senior debut during the series and is by far one of the youngest skaters in the Grand Prix.
Yūri!!! on Ice's storyboarder Kubo stated in an interview published in the October issue of PASH that Yuri is modeled after Russian figure skater Yulia Lipnitskaya. Currently eighteen years old, Lipnitskaya is also a young powerhouse skater and the youngest skater to ever win a gold at the Olympics at fifteen years old.
Victor Nikiforov and Evgeni Plushenko and John Cameron Mitchell
Victor, winner of five consecutive World Championships, Grand Prix, and many other competitions, is the rockstar-esque and legendary retired Russian figure skater who decided to coach Yūri. With that record, isn’t difficult to find discern Victor's potential real life counterpart.
Plushenko, four-time Olympic medalist, three-time World Champion, and four-time Grand Prix Final champion, is a legend with an attitude in his own right. Notable as well is how Victor rests his forefinger like he is signaling &quot;hush&quot; that appears to be a nod to Plushenko hush of the crowd in Sochi 2014 Olympics. However, Victor as Plushenko could be up for debate. According to a tweet from Victor's voice actor Junichi Suwabe, Plushenko is not a model for Victor. Is Victor so legendary no one in real life can touch him?
Victor is also partially based on American actor John Cameron Mitchell. In a tweet from Kubo, when she first thought about Victor’s visual design last year, she thought of Mitchell, an actor she saw during a trip to New York.
Yūri Katsuki and ???
Who is Yūri's real life counterpart? According to Kubo, Yūri is not merely one Japanese figure skater. Rather, she wanted him to be a character that could reflect all Japanese figure skaters and also be appreciated as a character on his own.
However, it is easy to find bits and pieces of other skaters in Yūri. As seen in numerous fan made videos, Yūri's short program Eros matches Japanese figure skater Yuzuru Hanyū 2014 Sochi Olympic Short Program. Beyond Hanyū, figure skater Tatsuki Machida influenced the series and potentially Yūri. The retirement of figure skater Tatsuki Machida's retirement partially inspired Kubo to create the series and make it successful.
(Emphasis mine)

From Topic Station (Japanese), regarding Yuri Plisetsky's model,

When Kubo was interviewed regarding Yuri Plisetsky's voice actor, she mentioned a little bit about him,

Why Russian athletes are difficult to approach, it's because the appeal(?) is great. When I saw (Yulia) Lipnitskaya, I thought &quot;If she was a man, it'd be the best&quot;, then directly used her as Yuri's image.



Nobunari Oda, a Japanese figure skater, requested to participate in the anime. He appears as a commentator in episode 11 and voiced his own character.
From Oricon News (Japanese), regarding Nobunari Oda,

Oda, who watched the anime every week from episode 1, quickly pleaded, &quot;I want to appear as myself and commentating Yuuri's skate program even only for once!&quot;

 

Stéphane Lambiel, a French commentator in episode 12, was also a real-life figure skater. While he was born in Switzerland, he also speaks French, English, German, and Portugal. His outfit was also traced from 2016 NHK Trophy.