The authors give \(L^p\)-estimates for stochastic integrals of the C. Barnett, R. F. Streater and I. F. Wilde type [J. Funct. Anal. 48, 172-212 (1982; Zbl 0492.46051)]. These inequalities enable the extension of the stochastic integrals to all \(L^p\)-spaces for \(1\leq p < \infty\). The integrals are taken with respect to a pair \(P, Q\) of independent free fermionic fields. Let, for a simple adapted integrand \(G\), \(\langle G, G\rangle=\sum_{j}G_{j}^*G_{j}(t_{j}-s_{j})\), and let \(\langle{\mathbb G}, {\mathbb G}\rangle\)= \(\langle G_P, G_P\rangle + \langle G_Q, G_Q \rangle\). Then, for \(1\leq p \leq 2\), there is a constant \(K_p\) depending only on \(p\) such that \[ \|G_P dP\|_p + \|G_Q dQ\|_p \leq K_p\min\{\|\langle {\mathbb G},{\mathbb G}\rangle^{1/2}\|_p, \|\langle {\mathbb G^*},{\mathbb G^*}\rangle^{1/2}\|_p\} \] for any pair \(G_P, G_Q\) of simple adapted processes. Moreover, an \(L^p\)-martingale with \(1<p\leq 2\) can be written as a sum of stochastic integrals with respect to \(dP\) and \(dQ\), and the corresponding integrands \(G_P,G_Q\) are given by an explicit formula. To this end, a differential calculus for the Clifford algebra has been developed. This work should be compared with the paper by G. Pisier and Q. Xu [Commun. Math. Phys. 189, No. 3, 667-698 (1997; Zbl 0898.46056)].