Yes. If you slip and fall in a supermarket and sustain an injury, then it is possible to sue the supermarket for compensation for the injury you have sustained. This is known as an Occupier’s Liability Claim or more broadly a Public Liability Claim. The supermarket will have Public Liability insurance which will respond to any claim you make for injuries sustained if you slip and fall in a supermarket.
To obtain any compensation, you must be able to establish how you fell and what caused you to fall. You will only be successful in a slip and fall claim against a supermarket if you can prove negligence. Just because you fell and sustained an injury does not mean that you will win your claim. You must establish breach of duty of care. Breach of duty of care is a failure by act or omission on behalf of the supermarket to have a reasonable system of inspection and cleaning in place to pick up and clean any spills on the floor so people do not slip.