As far as I know you can not stay logged into PBase on you iPhone. However you can enable autofill for Safari, thus reducing the keystrokes to log in. Just search for "autofill iPhone" there are several films how to do this. Here is one: https://www.youtube.com/watch?v=_vj2vJA3kpA
Interesting. Unfortunately I don't have an answer for you, but I stay signed in with my iPhone and iPad and computer. In fact it's been so long since I entered a password I'd have to look it up. I am wondering if there is a Keychain setting to stay signed in? Or maybe Safari settings for password and autofill? I briefly looked and didn't find anything that says stay signed in. So I am baffled, but I do know the iPhone can keep you signed in.
Yes, it's weird, my iPhone appears to stay logged in and I can't at this point recall what I did when I first logged in. I don't think too much about it, as I don't find the phone to be a very happy interface to PBase, but I can check my stats and look around a bit whilst on the road.