Bull-Rushing someone off a cliff

Can you bull-rush someone off a cliff, with either melee or a spell such as hydraulic push?

Bull rush says You cannot bull rush a creature into a square that is occupied by a solid object or obstacle. Does an obstacle include said cliff?

If they can be pushed off would they get any kind of save?

Judging by the wording I would say yes:

You cannot bull rush a creature into a square that is occupied by a solid object or obstacle.

A cliff (down) does not occupy the square they are being pushed to. It is in fact a lack of anything in that square, and in the squares theoretically below it. In no way does it obstruct your movement into that square, it just causes you to keep moving once there :-).

The climb skill specifies the check for catching yourself when falling:

Catch Yourself When Falling: It's practically impossible to catch yourself on a wall while falling, yet if you wish to attempt such a difficult task, you can make a Climb check (DC = wall's DC + 20) to do so. It's much easier to catch yourself on a slope (DC = slope's DC + 10).

In the case where a character is bull rushed more than one square away from the edge, I probably wouldn't even allow the climb check (there's nothing to hold on to).

In these posts on the Paizo forums the assumption certainly seems to be that it can be used to push an enemy off a cliff, but there is no official backing.

The following post asks the same question, but gets no official response.

There are no invisible "video game" walls in an RPG, unless you put them there.

Doing so safely may not be an option for most characters; however, saftey wasn't the question.

Mid-air is counted as "unoccupied" for the purpose of determining whether or not you may move into a space.

To further illustrate this point, when you make a jump check, you have to jump into the air, and any character can most clearly jump off of a cliff at their own discretion. Inadvisable for most, but their own discretion is the only limiter in this case.

If you wanted to bull rush them INTO the BASE of the cliff, the answer is clearly "No", as the cliff base clearly occupies those squares.

Bull rushing them UP or DOWN the side of the cliff, while tricky, could theorhetically be possible, as you are pushing them into unoccupied squares on difficult terrain, but at some point the angle of slope would determine whether or not it would be possible, and/or require a climb check to complete if you have enough of a turn remaining to complete both actions.

As to the save, a previous answerer mentioned making a Climb check to save from falling. SevenSidedDice also pointed out the use of a Reflex Save, which more directly answers your question, as being appropriate.

ob·sta·cle/ˈäbstəkəl/ Noun:
A thing that blocks one's way or prevents or hinders progress.

quoted text from post (emphasis mine):

Bull rush says You cannot bull rush a creature into a square that is occupied by a solid object or obstacle. Does an obstacle include said cliff?

I interpret this to mean obstacle that hinders you from pushing them. An edge of a cliff surely does not qualify, in fact it helps you because the person you are pushing has no footing. It might prevent you from pushing them more than one square, but I am not going to address that.

also:

do pits occupy the spaces above them? I would imagine not... so the empty square above the pit that you push them into before they fall would not have an obstacle.

• +1 for noting that you can't push someone while standing over a pit unless you're Wile E. Coyote Jun 23 '12 at 5:20

If it were at my table I would rule that they could be pushed off of a ledge/cliff/etc. Otherwise there would be no mechanic for an otherwise very legitimate tactic. Anything else would be a house rule or a special sequence.