Falling is the logical conclusion from the associated rules
A literal interpretation of the Trip CM and Prone Condition suggests that yes, a climbing creature can be Tripped causing them to become prone on the floor under their current space
If your attack exceeds the target’s CMD, the target is knocked prone.
Prone: The character is lying on the ground...
Furthermore, this seems like a logical and fair result of the interaction. However, you may wish to take into account the result of taking damage allows you to make an Athletics check to not fall, although it is not directly related.
Anytime you take damage while climbing, make a Climb check against the DC of the slope or wall. Failure means you fall from your current height and sustain the appropriate falling damage.
Reducing or removing fall damage for the encounter is a reasonable modification, as the GM, if you expect to be knocking a lot of PC's down from the ceiling. Be sure to balance the penalty of losing time fighting on the ceiling and fall damage vs maintaining enough tension for the fight.
Also make sure to go over the particulars of the Climb skill before running an encounter heavy with it, such as the ability for characters to catch themselves (typical DC is 20 over the regular Climb DC, or 10 over on a slope) and that, even with easy Climb DC's, it's nearly impossible to fight in melee while climbing.
You need both hands free to climb, but you may cling to a wall with one hand while you cast a spell or take some other action that requires only one hand.
You may want to provide a scroll of Spider Climb or something similar if you have any PC's that this would cause problems for.
If it helps your consideration, Paizo codified it to be official in Pathfinder's 2nd edition.
If you would be knocked prone while you're Climbing or Flying, you fall (see Falling for the rules on falling). You can't be knocked prone when Swimming.
