We prove strong convergence of an iterative scheme for approximation of fixed point of $\lambda$-strict pseudocontractive mapping in a uniformly smooth real Banach space (which is not necessarily uniformly convex). We apply our result to approximation of common fixed point of a finite family of strictly pseudocontractive mappings. Our result extends the results of Li and Yao [M. Li, Y. Yao, Strong convergence of an iterative algorithm for $\lambda$-strictly pseudocontractive mappings in Hilbert spaces, An. St. Univ. Ovidius Constanta 18 (2010), 219-228] and complements other new interesting results in the literature.