Is there a way to keep 1Password unlocked on iOS?

From the User Guide:

Can I configure 1Password to never prompt me again for the password/code?

Yes, you should be able to but there’s a catch. This only works after you:

1. Initially unlock 1Password app already,
2. Set both Request After and Request Code After to "Never", and
3. Also keep re-opening 1Password once in a while.

Why the last one? When you stop using 1Password, the iOS app will keep track of how long it’s been in the background in order to ensure there’s enough memory for your current and most recently used apps. 1Password can remain unlocked in memory only until iOS is forced to fully close the app in the background to reclaim its memory block, so that your current apps can use it.

If the termination has occurred, you’ll have to enter the master password to unlock it.

Note that the more memory you have in your iOS devices, the longer 1Password can remain unlocked. Keep this in mind when switching to older iOS devices that might see 1Password locked more often.