The short answer is yes, it is possible. However, there are a few extra important things to consider.
The main thing is how long you will be able to run your Aircon for. Let us take one of the more popular and energy efficient Caravan Aircon models on the market: the Dometic Ibis4 Rooftop Aircon: The peak power draw is around 1400w to 1500w. So, you could use a quality 2000w+ inverter (note that Caravan Aircons are 240V) with say a 200ah lithium battery and you will get around 1-2 hours Aircon run time, before you must recharge the batteries again. Depending on how you recharge your batteries (DC to DC charger, solar, mains power etc.), this charging will take time.
So once again; you can do it with certain batteries & inverters if you are aware of how that will affect your battery storage and charge loss... and knowing how to manage that system.
You can get a larger lithium battery system to run your Aircon for longer, but obviously this will cost more, so that needs to be considered. It is certainly a big recent trend that our RV customers are installing quality lithium battery systems to run their Caravans/Motorhomes etc. However, running an Aircon unit can be one of the larger power drawing appliances in your van, so it pays to consider your power system's capabilities and how long you want to run the Aircon.
If you are just pulling up for short stops (1-2hrs) and you are doing a lot of driving, you can run your Caravan Aircon from your lithium batteries for this short time. Many RV owners do this, and as new Caravan Aircon models become increasingly energy efficient and lithium battery advancements evolve to be cheaper, more powerful, and even lighter, then we will certainly see this trend increase. However, currently the popular and often cost-effective choice to run your Aircon if you do not have access to mains power (i.e., free camping) is to use a small inverter generator. You can get a quality 2000-2400w sized inverter generator from around $1,000-$1,500 and these will run many RV Aircon units for 7+ hours before having to refuel.
So once again, you can use a battery system, but it is not always the most practical option. For more advice on Power Systems to run your Caravan/RV, feel free to give us a call on: 1800 787 278.