Is there any exchange where I can borrow fiat (usd or eur) against BTC? I know that Bitfinex offers swap market, however, you can use the money only for trading. I am looking for fiat withdrawal.
No, there are no service like that, all of exchangers gives margin trading not for withdrawal to fiat
the fetures is for trading on there, i think the best ways to fix your problem is Banks only.
You can visit the local Banks on your country and make loans fiat money,
don't waste your time to looking for the service that you want on exchangers of bitcoins.