The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points \( (k, m) p^{-m} \) with \( 0 \leq k \leq n < p^m \) and \( \binom{n}{k} \equiv a \pmod{p^s} \) (for \( (a, p) = 1 \)) for \( m \to \infty \) tends to the Hausdorff measure on the "\( p \)-adic Sierpiński gasket," a fractal set studied earlier by von Haeseler, Peitgen, and Skordev.