There may well be asbestos in the attic used for insulation usually the deteration is leaked out thru air vents located in attic. See that the celing is repaired pronto, as for your exposure to you and your eye don’t think that small amount just one time will harm. Please don’t worry yourself.

If the ceiling is made out of plaster it is more likely to have asbestos in it than if it made out of sheetrock. If it is a flat ceiling with no texture on it (such as a popcorn ceiling) and made out of sheetrock, it is 90% likely that it is not asbestos. Sometimes you can find asbestos in the joint compound that is used on sheetrock ceilings, and textured ceilings often have asbestos in them. However, even if it is asbestos plaster or sheetrock, getting a drop of water in your eye will not give you cancer. Asbestos is a hazard when it is airborne and you breath it in. Even then it can take up to 10 years for symtoms to show up. The chances of a small leak in your ceiling creating an airborne hazzard is small. Also like smoking, the more asbestos you breath in the more likely you are to get an asbestos related illness, this is called dose response. So again, even if the leak broke apart a very small piece of your ceiling your exposure would be so small that it is not very likely that you will get cancer or an asbestos related disease. If you want to know if material is asbestos containing, contact a local environmental consulting firm and have them come out and test the material. They will take a few small pieces of the ceiling and send it off to a lab. Then you will know for sure.

As far as your question about if all houses have asbestos, most older homes may have asbestos in the form of linoleum floors, exteior shingles, textured ceilings, joint compound, pipe insulation, and floor tile. Usually not all of the materials listed above are asbestos but all have the potential to be asbestos containing. You can still buy asbestos containing floor tile today, but not pipe insulation. I am an asbestos inspector in NC and statistically I have found that newer homes do not have asbestos in them but it is always possible.

The other answers explain how you can find whether there is asbestos in your ceiling.
About how you get cancer from asbestos:
Cancer from asbestos requires a prolonged exposure, not a single splash. (I don’t think that water had any asbestos, but even if it did). Usually asbestos requires inhaling it. All of us inhale about 10 000 to 15 000 asbestos fibers a day -without any so called exposure.So you need a lot more for months or years -in addition to what we all get. Already In ancient Rome they knew that slaves form asbestos mines die young -can you imagine the exposure those poor guys had had! so be of good cheer.If you want more exact data look in the national cancer Insititutes web pages and type in search: cancer prevention and asbestos. (www.cancer.gov)
Worry free spring
wishes mary a

There probably is no asbestos. Ceilings that look like cottage cheese with sparkles and if built before about 1974 , did contain asbestos. Most of these have been encapsulated (fancy word for painted) and are considered safe. Of course, if something were to hit the ceiling causing it to flake, would be a different story. A simple fix in this case would be to just repaint taking care not to saturate the material too much.

Most older homes have asbestos of some kind. If the home has a wall furnace, the piping that goes through the attic space and vents outdoors is called transite pipe and contains asbestos. Old 9X9 floor tiles that are adhered to the sub floor or concrete with a glue that contains asbestos.

If you need more assurances the home is safe, request the landlord have the house tested for asbestos.