Yes you do get a loan for used cars too. However, between new car and used cars the interest rates are higher for used cars. Normally this difference is around 3-4 %.
All banks offer used cars loans, these are ICICI, HDFC, Kotak to name a few from private sector, You can also try some of the NBFC's like: Tata Capital for used car loans.
The best and cheapest way to get a used car loan is to buy a used car from car manufacturer used car division like: Maruti True Value, Tata Assured, Hyundai Advantage and like-wise. Mahindra First Choice is also a good option for used cars.
Even your local used car dealer from unorganised market could get you a used car loan. We would however suggest arranging one yourself, even though it could be time consuming.
Before buying a used car, ensure that it had not met with any head-on collision, as such cars are really a lemon. This you can find from the insurance company by checking if there has been a big body repair bill for the car. You can also use services of your local trustworthy service or body repair mechanic.
In terms for formalities, once again these would be far lesser when you buy from some of the above mentioned car manufacturer driven used car division. Formalities in addition to identity and credit worthiness establishing documents, you would also have to submit some RTO mandatory forms for the loan purpose.
Identity proof can be through your residence address registration, electricity or water bill and ration card etc. Credit worthiness is through your form 16, pan card and bank statement. Car documents like : NOC from a bank if the seller had bought the car on Loan, Insurance paper without the bank's hypothetication. Car's registration and RTO paper to establish seller's ownership of the car.
Do also get a certificate from RTO department which clears any doubt about the car being a stolen property, which would not only make it a lemon for you not but can also get you into legal hassles.