Drake & Hailey Baldwin: First Date Pic

Drake extended his holiday weekend with Hailey Baldwin by getting cozy with her on a date ... the night after they hung out at his insane house party.

Our sources say Drake and Hailey slipped into The Nice Guy Tuesday around 11:30 PM and got a table with another couple. We're told the table ordered shots -- and while you can see Hailey spent some time staring into her phone ... she and Drake were cozy enough that it was clear they were on a date.

Drake even pulled the move -- y'know ... arm behind her on the sofa. More date evidence? They pulled that other move when they left around 2:30 AM -- she went out the front, Drake took the back door.

This comes on the heels of multiple reports they kicked it at his Memorial Day party, but now we have photo evidence.

Hailey's got a type ... rich, famous, singing Canadians.