The pickaxe is a new addition to Sons of the Forest and can be found in the new cave that was added with the latest update. As with most items found in caves, you'll need to venture pretty far in to find this item, and you'll need a rebreather to reach it too, so make sure you come prepared.
You might be wondering exactly what you need a pickaxe for when there's nothing to mine on the cannibal-infested island, but that isn't strictly true. The update also introduces Solafite, a resource needed to upgrade weapons, so naturally, you'll need a pickaxe to mine it. With that in mind, here's the Sons of the Forest pickaxe location, as well as where to find the new cave.
Sons of the Forest pickaxe location
You'll need to locate the new cave over in the northeastern corner of the island if you want to find the pickaxe. There are two small lakes connected by a river and the cave is found just to the north of the larger of the two. You can check out the exact location on the map above.
The cave entrance is more of a hole, but luckily you'll find a rope conveniently tied there, allowing you to climb down inside. Once you drop down through the eerily long tunnel, you'll arrive on solid ground. You'll need a light source to see where to go next—a lighter works fine if that's all you have.
Follow the tunnel until you come to an underground stream, then follow the water until you come to a drop. Use the rocks on your right to make your way down to the pool. This is where you need the rebreather, so equip it and jump in. The tunnel is long but linear, so follow it until you reach a larger cave.
From here, head the only way you can. There's a boarded-up doorway to your left which you can break through to grab the furnace recipe. Back outside, go right and follow the tunnel until you reach a water slide. You'll need to go for another swim once you reach the bottom, so dive and follow the narrow tunnel until you reach an area where you can surface. There's a narrow place to climb out of the water with a broken board fence nearby. Pass through here and dive again into the next pool.
You'll be greeted by several hanging dead men when you get out of the water on the other side. The sight might not fill you with hope but you're almost there—head towards the light source on your left, and you'll find the pickaxe stuck in a Solafite deposit next to the step ladder. Once you've got your prize, you can get to mining straight away, or, if you'd rather hurry out of the cave, you can follow the tunnel forward and you'll eventually find a series of ropes to climb that take you back to the surface. Phew!