For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.
The paper also shows how to generate certain other distributions on the $\ell_p^n$ ball by modifying the distribution of $Y$.
For arbitrary p, this paper does exactly what you want. Specifically, pick $X_1,\ldots,X_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector $$\frac{(X_1,\ldots,X_n)}{(Y+\sum |X_i|^p)^{1/p}}$$ is uniformly distributed in the unit ball of $\ell_p^n$.