Let { X n k , 1 ≤ k ≤ n , n ≤ 1 } be a triangular array of row-wise exchangeable random elements in a separable Banach space. The almost sure convergence of n − 1 / p ∑ k = 1 n X n k , 1 ≤ p < 2 , is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in(Rademacher) type p separable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.