The Gunners were looking for their first Champions League title but suffered heartbreak in Budapest as they were beaten 4-3 on penalties by PSG after a tense 1-1 draw in 120 minutes, Eberechi Eze and Gabriel missing from the spot.
Mikel Arteta's side therefore end the season with only the Premier League trophy but Rice insists there is more to come on the silverware front in the future.
Speaking to TNT Sports, Rice said: "The emotions and the stakes are so high. It's cruel. He [Arteta] spoke about how much he loves us as a group. How we've given 100 per cent every single game with everything that's been thrown at us.
"This is only the start for us. We got over the line in the Premier League, this would've been one step further but it wasn't to be. We keep building. Since I came to the club it was a quarter-final exit, then semi-final and now final. We keep going and keep staying positive. This isn't going to define us."
- PSG 1-1 (4-3 on pens) Arsenal - Match report & reaction
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Rice: We had a similar penalty against Leverkusen
The one moment Arsenal fans may look back on with disappointment was the decision not to award a penalty in extra-time when Noni Madueke went down in the area under a challenge by Nuno Mendes.
Rice and Arteta both had their say on the incident at full-time after being booked for their remonstrations.
"I've not seen it back yet, but at the time on the pitch I thought Noni was ahead of him," said Rice.
"Looking back at the one we got against Bayer Leverkusen, it was very similar.
"I'm not going get into it now. At the time I was devastated. Against PSG, you have to be mentally on it. I thought as the game went on we grew into it and created a number of chances.
"PSG have lost loads over the years and now they are having their time. I think that's going to keep happening with us. We are having our small losses but we are going to go on and keep pushing."
Arteta said: "I watched it back and it can easily be a penalty. Basically we see the penalties they have given in the competition this season. The referee's made that decision but he made a different one with Mosquera. That's an important one."
What's next?
Already thinking about the 2026/27 Premier League season?
There's a World Cup to fit in first - follow live blogs of every game on the Sky Sports App - but the key dates are locked in for next term, when Sky Sports will once again show at least 215 live Premier League games.
Here's your summer diary:
June 11: World Cup starts - day-by-day fixture list
June 14: Scotland's opener vs Haiti (2am)
June 15: Summer transfer window opens
June 17: England's opener vs Croatia (9pm)
June 19: Premier League fixtures released (10am)
July 19: World Cup final
August 22-23: Premier League opening weekend
August 27: Champions League league phase draw
September 1: Transfer Deadline Day (window closes at 11pm UK time)
September 8-10: Champions League matchday 1
May 30, 2027: Premier League final day
