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If you are injured in a store parking lot and it is due to the store's negligence, you can sue. Keeping the parking lot safe is part of the business's responsibility under premises liability law.
But not all injuries arise from negligence. In order for the store to be responsible for your injury, the store must have failed in its duty. That failure need not be direct but the cause of injury can't be disconnected or impossible for anyone to anticipate.
Premises liability law provides that owners are responsible for accidents and injuries that occur on their property. That means if you slip and fall inside a store or out in the parking lot, and the store failed to make the space safe, it will have to pay for medical expenses, lost work time, and maybe more.
Premises owners have a duty to maintain their property, so even if they do not know about a particular danger and they should have, they can be held liable for an injury. Say you slip in the store parking lot and fall because it is iced over, your lawyer must show that the management knew or should have known, based on weather reports and usual conditions in the parking lot, that the area would become slippery.
If the store fails in its duty to maintain the property and is shown to be the cause of an injury, then your claim succeeds. But sometimes things happen that no one can anticipate.
If a snow storm hits Miami in July and that is the first time in the history of weather that such a thing has happened, the store is probably not liable for failing to anticipate the need to keep its lot ice-free.Your slip and fall claim is unlikely to succed. The same cannot be said for a store in New York in December.
Every case is made on the details, which is why it is impossible to say whether you can sue from an injury that occurred in a store parking lot without knowing more about what happened. Talk to a lawyer. Many attorneys will consult for free and some will take your case on contingency, meaning you will not pay a fee unless they can win a damages award for you.