Hmmm... nobody has ever asked that before. If you could call the Wilderness Permit Office at 760-873-2483, that would be the final word. Please let us know what you find out.
I am thinking they would not require one for a baby.
By the way, you only need a permit if you go beyond Lone Pine Lake. And since it is so early in the season, you are likely able to go into the Visitor Center and pick up a free walk-in permit.
