The Gunners have never been crowned champions of Europe but came closer than ever before in Budapest.
Kai Havertz gave the Premier League winners an early lead, but Ousmane Dembele's second-half penalty brought PSG level.
The French side prevailed 4-3 on penalties to retain the trophy after Eberechi Eze and Gabriel Magalhaes failed to hit the target with their spot-kicks.
"I think we've come really far and this is only the start for this group," England midfielder Rice told TNT Sports.
"Obviously we got over the line in the Premier League, which was a dream come true, this would have been one step further.
"It wasn't meant to be, but we keep building. Since I've come to the club, it was a (Champions League) quarter-final exit, semi final, now a final, so look we keep going, we keep staying positive."
PSG struggled for years to get over the line in the Champions League despite huge investment from their Qatari owners.
But they have flipped the script under Luis Enrique to become only the second side in the Champions League era to defend the trophy.
Rice said that is an example Arsenal can follow.
"They've lost loads over the years and obviously now they're having their time," he added.
"I feel like that's going to keep happening with us.
"We have these small losses, but we're going to go on and keep pushing to try and achieve something big."