Are single control faucets supposed to leak when pressed down?
No, the faucet should not leak like you describe when moderate pressure is applied directly straight into the faucet knob. Excessive force there is a possibility that it would leak unless you broke it first in which case you would likely have a gusher to deal with.

There are seals onside this type of faucet assembly that can get worn, old and hard and even crack with age and use. When the seals start to fail it can leak in various ways under multiple use scenarios. Sometimes it is simply a process that the seals need some lubricated with food grade petroleum jelly. But in most leak situations like you describe you would have to turn off the water source and take the faucet apart so that the seals and maybe even the ball valve can be replaced.