Is an interest-based loan permissible in some cases?

Necessities make forbidden things permissible. This includes riba (usury/interest), whose perpetrator Allah and His Messenger have declared war against. As Allah Almighty says: "And He has already explained to you what He has forbidden to you, except that to which you are compelled," and His saying: "But whoever is compelled [by necessity], neither desiring [it] nor transgressing [its limit], there is no sin upon him. Indeed, Allah is Forgiving and Merciful."

The compulsion here is for the borrower, not the lender, and it has conditions, including: 1. The necessity must be present, not anticipated. 2. There must be no other means to دفع (ward off/address) the necessity except by violating the lawful commands and prohibitions. 3. The compelled person must observe the extent of the necessity, because what is permitted due to necessity is measured by its extent. 4. The compelled person must not embark on an action for which a dispensation is never permissible.

The compelled person should remember Allah Almighty's saying: "And whoever fears Allah - He will make for him a way out," and "Indeed, with hardship [will be] ease," and "Or who responds to the desperate one when he calls upon Him and removes evil."