SOME .22lr ammo does lead up.
Cheap ammo is more likely, but not always.
Better grade ammo, especially the plated bullets like Remington's "Golden" bullets don't unless you have a barrel problem.
I once had a 5000 round case of Remington standard velocity un-plated that leaded so badly that it would actually almost close the bore off.
I discovered this when shooting a Calico carbine with a 100 round drum magazine.
I was shootin' along having fun when I noticed that as I fired a shot, SOMETHING hit the berm BELOW and to the side of the target.
I started watching the shots through my spotting scope and wasn't seeing new holes in the target but more strikes on the berm.
A look down the bore and I was shocked to see a pin hole instead of a clear bore. How the bullets were getting out, I don't know, and fortunately, the barrel wasn't damaged.
The same ammo leaded EVERYTHING I shot it in, from revolvers to a Remington 40X match rifle.
I sent the remaining 4000 rounds back to Remington, and they sent me new ammo, along with a letter saying that they found that ammo lot had had a problem with improper lubrication during manufacture.
Generally speaking, the only time you see much leading is in the forcing cone area of revolvers, or in bad barrels.