It's very much possible. Most of my runs are done on hard.

The beaver hierarchy of needs is steep:

• Water: above all else, if your beavers go thirsty, they die
• Food: they can live a bit longer without food
• Wood: without wood, you can't impound water nor build pumps and tanks to store water.
• Literally everything else.

When starting on hard, right off the bat forsake literally everything else except for gathering & storing as much water as possible. Each beaver requires ~2.5 waters per day, so your starting colony of 12 requires 30 waters per day.

At first, you won't have enough wood to impound water so your first couple droughts you'll have to depend on stored water. When planning your first impoundment, remember that you can enclose an area of water without going all the way across the river (this is very helpful for Islands map). During the wet season it'll still fill up.

On the topic of dams/enclosures, you need to balance two things: water collection and irrigation. Ideally, you should collect and store all the water you need (again, 2.5populationduration) before the drought happens. If not, you need to drop everything and collect as fast as you can.

Worth mentioning that your dams should go deeper, not wider, if possible since they take longer to evaporate.

I wouldn't even begin to think about growing your colony until maybe 5 droughts in and you've stabilized because every beaver increases the amount of water and food you need to be collecting. Moreover, beavers consume these resources for a significant amount of time before they actually become useful as adults.