Is there any evidence in the Prophet's Sunnah or the sayings of the Companions that indicates the virtue of walking specifically as a form of exercise?

No saying has been reported from the Prophet, peace and blessings be upon him, regarding walking as a form of exercise, nor from any of his Companions. However, it is reported that he raced 'Aishah, may Allah be pleased with her. When the Prophet, peace and blessings be upon him, walked, he would incline forward, and he was the fastest and best walker among people. He also jogged (khabb) during three circuits of his circumambulation (tawaf (definition)), and jogging is a type of swift walking. Among what is attributed to Ali, may Allah be pleased with him, is that he ordered walking after Isha prayer, but the correct view is that this statement belongs to Al-Harith ibn Kalada. 'Umar raced people and outran them, then Ibn Al-Zubair outran him. Salamah ibn Al-Akwa' was permitted to race a man, and Ibn Al-Akwa' outran him. In the Hadith (definition): "Travel so that you may be healthy," due to the exercise involved therein. Physicians have mentioned that walking provides exercise, strength, and metabolism, and that slightly tiring the body is among the things that preserve health.