Police closed Marine Road Central after the crash on the seafront at around 12.50pm.
Officers say the car collided with two pedestrians walking a dog near the King's Arms pub. The dog reportedly ran away from the scene after its owner was injured, but has since been found.
North West Ambulance Service said the pair have been taken to hospital. Details on their injuries are not known at this stage, but Lancashire Police described the collision as "serious".
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The force said a 24-year-old man has been arrested on suspicion of drug driving and is currently in custody.
Marine Road Central was closed both ways, between Northumberland Street and Pedder Street, but has since reopened.
Earlier today, a police spokesman said: "We are currently at the scene of a serious collision involving a car and two pedestrians Northumberland Street, Morecambe
"Emergency services have attended and the road is closed.
"We are advising motorists to avoid the area and will update you in due course."