Plumbing – How to remove the water/temperature control faucet if there are no visible screws

I have THE EXACT faucet. There is NO screw. JUST PULL UP! May have to do a slight pry on the base but that is what it is, no screws.

It looks like you have one of Delta's older 3530 models. Looking at the exploded diagram for this model shows part # H24 (the handle covers) with only a set screw holding it in place. If you've already removed that and the cover still won't lift off, it may be bound up on the stem from having been turned too far in one direction or the other (if it was leaking, I could imagine someone may have tried extra-hard to turn it off). I would try slowly but forcefully turning the faucet handle past its resistance point, with it in the on position first, then if that doesn't loosen it, try doing the same but the other direction with it in the off position. You may have to alternate this a few times to get it to free itself from the teeth on the stem.

If none of that works, there is a retaining nut on the underside of the sink counter (part # RP9519 in the diagram). It's not a very thick nut, so you'll have to put your hand right against the underside of the counter top to feel it. The threaded part of the stem assembly likely extends well past the nut, you'll need to make many turns to get it off. Taking the handle out from the counter will let you examine it in more detail and you can take tools to the less attractive parts to gain torque, hopefully avoiding any more scratches on the handle itself. Just be careful not to grip too tightly on the screw threads or you won't be able to reattach it to the counter!