I've been chased by cops on foot twice and stopped a couple of other times, nothing too exciting though. However, a friend of mine has a story that tops even the one dude's dad who dislocated his shoulder. I don't think he'll be on here to post it so I'll give my best shot:
4th of July, early '90's, Seward, Alaska. If you've ever been around there on the fourth you know what kind of party this is. In the 90's it was even crazier. On the fourth there are 40k people trying to leave, with one two-lane road going in and out of town. Traffic was blocked, so "Gary" decided he would get out and run naked past all of the cars, until traffic freed up and his driver caught up to him. Unfortunately, after about a mile of streaking he caught up to the traffic blockage, a car accident. It was surrounded by cops, so Gary had to turn around and run back past however many hundreds of cars that he had just flashed his ass at.
Reactions were mixed; some laughed and smiled, while others were horrified or covered their children's eyes as the streaker ran by. I should mention at this point that Gary is pretty fast, and was pretty keyed up at this point. He was running probably mid-5's, and somehow never saw his friend and the car. So he kept running, all the way back into Seward.
There were still plenty of people wandering through the streets of this extremely small town, and not all of them were sympathetic to poor Gary's predicament. After what he estimates to be about 3 or 4 miles, through streets alleys and the tent city on the beach (realizing much later that he should have dived into one of them and holed up for a while), all of the cops in the city were after the crazy naked guy tearing through the streets of Seward. There were lots of extra officers there for the fourth of July madness, so that meant some pretty creative evasion for Gary.
At some point he was getting pretty tired, with no sign of his friend anywhere. He ducked into a hotel and faced a shocked manager. With one hand covering his package, he confronted the man.
"I made a bad decision," Jerry said. "Can I hide out here for a few minutes?"
The hotel manager threw Jerry back out onto the street, and the cops were on his trail. He had a couple of them chasing him from behind, and another cop blocking the other end of the street. In the moment that followed, Gary made the crucial decision to run straight at the cop blocking his path. I like to imagine the cop's expression at the instant that a completely naked guy ran directly at him at full speed. At the last minute Gary faked to his left and cut hard to the right, as the cop faceplanted trying to tackle him.
At this point Gary had been streaking for at least half and hour and was pretty winded. With no sign of his friend he decided to give himself up, and spent a night in jail. I can't remember how many cops were actually after him, or were there when he was finally arrested, but I doubt anyone here can beat the number.