Jill Biden came ready to play -- and no, not with Joe Biden -- but for a very pricey TV cameo!

The former First Lady dropped a $35K bid for a walk-on role in the next season of "Heated Rivalry" at the NYC LGBT Community Center’s Center Dinner Thursday night ... but got iced out.

Jill's offer barely made a dent -- the role sold twice for $125K a pop -- and Jill later joked on social media she "won’t be heading to the cottage after all."

Still, if this proves anything ... it’s that "Heated Rivalry" has people in a full-on frenzy -- even Jill’s catching feelings!

The hockey hit follows the forbidden heat between Hudson Williams' Shane Hollander and Connor Storrie's Ilya Rozanov -- with fans so invested, they’re shipping the stars IRL. Season 2 is scheduled for April 2027.

As for Jill ... looks like she’ll be skating into Season 2 from the sidelines.