A turquoise Vauxhall Corsa, a white Ford Transit and a green DAF lorry collided at about 11.40am between the Ford Road roundabout and the Causeway roundabout close to Arundel railway station.
Long queues built up after the road had to be closed while emergency services worked at the scene.
A 57-year-old woman from Littlehampton, who was driving the car, was taken to Worthing Hospital for treatment for suspected pelvic injuries.
At one stage it was feared that she would need to be cut out of her vehicle but this did not prove necessary.
Her distressed puppy was removed safely from the vehicle and given to her family to care for.
A 36-year-old man from the Netherlands, who was driving the lorry, was arrested on suspicion of causing serious injury by dangerous driving.
He was taken into custody to be questioned with the aide of an interpreter.
PC David Craker said: “I’d like to speak to anyone who saw the crash or any of the vehicles involved being driven in the minutes before the collision.
“If you saw anything at all, no matter how small the detail, please contact us.”
Those who saw the collision are being urged to call police on 101 quoting serial 520 of 15/4 or email [email protected]