Why is ice forming under the bottom drawer of my Samsung American fridge freezer Rsa1shpn and then water eventually leaking out the door? I have defrosted the fridge freezer but still have the problem

Something here should cover your problem.
side by side refrigerator ice on freezer floor YouTube

clogged drain

If it is a frost free unit the water collects in a well at the compressor , there is a heater element in the well that boils off the water , this is were the fault is.

When a refrigerator leaks water like yours, it is common for the problem to be nothing more than a clogged defrost drain. When the unit defrosts itself, the frost melt water drips into a pan underneath the evaporator in the freezer, and goes through a tube from there to the bottom outside of the unit. Since the water cant go down this drain, it backs up on the freezer floor, then eventually the kitchen floor.

you could have cloged drain,check for frozen water in the bottom of freezer or if coming from ice maker.

Frost free fidges are supposed to stay frost free because the automatically defrost regularly. When they do this they may drain water down to the bottom of your fridge. Look for a tray that catches this water underneath the unit. Make sure yo keep it emptied. If you live in a very warm climate you may be getting a lot of humidity inside. There may be a switch that you can change for lower moisture.

My Bosch Frost Free Fridge Freezer did the same thing - Ice forming in the bottom of the freezer area after a few days.

It was caused by a blocked drain pipe - When the freezer does the de-frosting cycle, the melted ice drips into a tray and tehn runs into a dish on top of teh compressor, through a pipe.

If this pipe gets blocked, the de frosted ice overflows from the tray and overflows into the bottom of the freezer and then re-freezes !

To fix, pull the freezer away form the wall, locate the pipe that runs to the dish on top of the compressor and blow through it - This should clear it. Alternatively, from inside the freezer compartment, pull off the cover at the back, but the clips holding it on will probably break off, as mine did ! You can sort of push the cover back on and re-fit the food draws.

Hi, You will find that your problem relates to the defrost heater in the freezer. You will need to take the drawers out and remove the back panel to access the evaporator. There is a foil heater that lays underneath the evaporator and it will need to be changed. Along with the klixon that controls it. This will ensure that both parts are good. Your problem will then be fixed no more water!
Good luck................

Sounds like a blocked defrost drain. By "American" I presume you mean side-by-side.
I had a similar problem with a freezer-on-top style Samsung. The solution to that was to defrost the freezer, remove the false backing plate in the freezer (required removal of the ice cube tray. the panel just pops out with a bit of pulling). The drain hole beneath the freezer coils was blocked with ice. This drain passes down into the fridge and then out the back of the fridge into the evaporation pan next to the motor. There is a collector pan under the freezer coils, this and the drain hole were filled with ice. When the freezer goes into its self-defrost cycle the water collects here and then overflows into the fridge.
I removed the ice with hot water/vinegar mixture (so it doesn't re-freeze immediately) and simultaneosuly used a bit of chiselling with a screwdriver to clear the drain plug. I did this from inside the fridge working up the drain hole into the freezer. The whole job took about an hour.

probably the defrost drain blocked. remove back panel from inside freezer and melt ice off with hairdryer and follow up with hot water till water flows away freely.
you may find all the pipework iced up and if so you have a problem with defrost system.
an alternative way is to switch off for 24 hours and leave doors open