12 Volt Solar Air conditionair
Ice is a great way to store "coolth" (the opposite of warmth) because (1) the phase change from ice to water absorbs a lot of energy per unit volume without changing the temperature, (2) the melting temp of ice (32F) is a pretty good temperature for cooling and dehumidifying air for human comfort, and (3) as ice turns to water, the liquid flows so you can easily keep melting more and more ice by just circulating the water around it. 

Here is a product designed to move cooling loads from daytime (when air is hot and electricity is typically more expensive) to night...

http://www.ice-energy.com/

12 volt DC air conditioners are available for use on boats and RVs, although the ones I've seen are expensive and inefficient.

Sharp recently announced that it will be making DC-driven air conditioning units...

http://hardware.slashdot.org/story/15/07/30/0221235/sharp-announces-sales-of-dc-powered-air-conditioner-other-products-to-follow

...so hopefully options will improve soon.

As to weather you would be able to cool your 10x10 room overnight from a 900 watt solar panel, that depends on a lot of factors including the insulation value and the air tightness of the walls, what heat sources are going to be inside the room while you are trying to cool it, and the temperature and humidity of the outside air.

If, for example, you made a 10x10 room located in Seattle out of Vacuum Insulated Panels, and only wanted to keep a single bottle of beer in there, then I think it would definitely be possible. If you are trying to cool a 10x10 stick hut in Mumbai with a family inside, then almost certainly not. In between - maybe... :) 

Great question, BTW. I have been working on a project called Bed Box to bring the joy of air conditioned sleeping to the developing world. The Bed Box is basically a semi-insulated tent that you sleep inside and has a hyper-efficient and silent AC unit attached to it. By reducing the load on the system (mostly by reducing the volume to be cooled), it would be possible to operate a system like this on a small 900 watt panel, and by using a phase-changing gel pad inside the BedBox, you could store a lot of "coolth" during the day and then be able to hold much of the cool temp throughout the night.