The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space mathbb {R}^X) is denoted by C_p(X). In this paper we continue the systematic study of sequences spaces c_{0} and ell _{q} (for 0<qle infty ) with the topology induced from mathbb {R}^{mathbb {N}} (denoted by (c_{0})_p and (ell _{q})_{p}, respectively) and their role in the theory of C_p(X) spaces. For every infinite Tychonoff space X we construct a subspace F of C_p(X) that is isomorphic to (c_{0})_p; if X contains an infinite compact subset, then the copy F of (c_{0})_p is closed in C_p(X). It follows that C_p(X) contains a copy of (ell _{q})_{p} for every 0<qle infty . We prove that for any infinite compact space X the space C_p(X) contains no closed copy of (ell _{q})_{p} for qin (0, 1]cup {infty } and no complemented copy for 0<qle infty . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.