This is not an answer, but rather a counter to Micael Rize's answer. I don't intend for this to be picking on him, but it is too difficult to explain in a comment, and there are many arguments about his answer, so hopefully this will be persuasive.
In my example, I am going to use only 16 zebras, but it should be obvious that this is can extend to 100 zebras as well.
This is the starting configuration.
Now, lets say the lion chases the zebra to the east. After the first move, that zebra is the closest, so it will move directly away from the lion. Subsequent moves will bring the lion closer to the zebras adjacent to the zebra being chased, so they will have to move as well, always directly on a path from the origin according to the strategy. After a while (trillions of years), you will have the following setup.
Even though the lion looks close to the zebra, since the starting configuration was so immensely large, they are still light years apart and this zebra is in no danger of being eaten. The only reason the lion caught up is that the lion made many more moves than this zebra because the other zebras were forced to move at one point. Now, however, the zebra is able to move every time the lion moves, so can stay in front indefinitely.
In any event, this is where the lion changes tactics. Instead of chasing down the current zebra (pointless), he heads straight north. The zebra will continue to move outwards since it is still the closest. Eventually, you will get to the point where it is no longer the closest zebra, like so.
Now the lion turns back to the pack and chases the closest zebra in the circle (Red). According to the strategy, this zebra can only move outwards from the origin which gets closer to the lion obviously allowing the lion to capture it. The other option along this path is to move closer to the origin. While this is away from the lion, it will eventually meet and surpass the other zebra in red on the opposite side of the circle. Even if we ignore the fact that the other zebras will all move a bit as the lion runs through the centre of the circle, the lion is now chasing two zebras. According to the strategy, they can only move away from the origin, so they will always stay together. Thus, the lion is twice as fast as them and will catch one eventually.
There are probably different strategies that this zebra can take to avoid capture (e.g. taking different paths not directly from the origin), but the current answer is not sufficient to explain how to stay away from the lion.