Hope I’m not stepping on anyone’s toes here but I would like to suggest a hard reboot of your iPhone
@Auri
Here is a link to instructions if you need them.
https://m.wikihow.com/Hard-Reset-an-iPhone
Before you do the hard reset I would suggest you repeat the cache and cookie clearing you were asked to do before. iPhones and iPads are kind of fussy creatures and a hard reboot sometimes helps.
It sounds to me as I follow this thread that the problem can be isolated to your phone. I don’t know if you have tried to log in from a laptop or other non iOS device to confirm that. If you have not, and if you can it would help to isolate the problem to your phone or to your SF account.
Failing that, I would suggest a factory reset of your phone. Back up everything first of course.