Let \((X,\|.\|_X),(Y,\|.\|_Y)\) be Banach spaces with norms \(\|.\|_X\), \(\|.\|_Y\), and let \((S,{\mathcal S},v)\), \((T,{\mathcal T},\mu)\), be \(\sigma\)-finite measure spaces. If \(1\leq p \alpha\})= 0\}. \] The class of bounded linear operators from \(X\) into \(Y\) is denoted by \({\mathcal B}(X,Y)\), and the adjoint space of bounded linear functionals on \(X\) is denoted by \(X^*\). The results of this paper relate to estimates for integral operators with operator-valued kernels of the form \[ {\mathcal K}(f)(t)= \int_S k(t,s) f(s)\,dv(s),\;f\in L^p(S,X), \] where \(k: T\times S\to{\mathcal B}(X,Y)\). In particular, in accordance with estimates known to be applicable to cases in which \(k\) is a scalar-valued kernel, it is shown that: if (i) \(\sup_{s\in S}\int_T\| k(t,s)\,x\|_Y d\mu(t)\leq c_0\| x\|_X\) for \(x\in X\), (ii) \(\sup_{t\in T}\{\int_S\| k^*(t,s)\,y\|_{X^*} dv(s)\leq c_1\| y\|_{Y^*}\) for \(y\in Y^*\), then for \(1\leq p\leq\infty\), \({\mathcal K}: L^p(S,X)\to L^p(T,Y)\), and \[ \|{\mathcal K}(f)\|_{p,(T,Y)}\leq c^{1/p}_0 \tau c^{1-(1/p)}_1\| f\|_{p,(S,X)}. \]