This paper is concerned with the boundedness of pseudodifferential operators \(a(D)\) with operator-valued symbols \(a(\cdot)\). Let \(E\), \(E_0\), \(E_1\), \(E_2\) be Banach spaces such that a multiplication \[ E_1\times E_2\to E_0,\quad (e_1,e_2)\mapsto e_1\cdot e_2 \] is defined and a continuous bilinear map of norm at most 1. It is shown that \[ (a\mapsto a(D))\in{\mathcal L}(S^m(\mathbb{R}^n, E_1),\;{\mathcal L}({\mathcal B}^{s+m}_{p,q}(\mathbb{R}^n, E_2),{\mathcal B}^s_{p, q}(\mathbb{R}^n,E_0))), \] where \({\mathcal B}_{p,q}(\mathbb{R}^n, E)\) is a Besov space of functions with values in \(E\) and \(S^m(\mathbb{R}^n, E)\) is the set of \(E\)-valued functions satisfying \[ |\partial^\alpha a(\xi)|_E\leq C(1+|\xi|)^{m- |\alpha|},\quad |\alpha|\leq n+1, \] and that \[ (a\mapsto a(D))\in{\mathcal L}({\mathcal F} L_1(\mathbb{R}^n, E_1),\;{\mathcal L}({\mathcal B}^s_{p, q}(\mathbb{R}^n, E_2),\;{\mathcal B}^s_{p, q}(\mathbb{R}^n, E_0))), \] where \({\mathcal F} L_1(\mathbb{R}^n, E)\) is the set of Fourier transforms of functions belonging to \(L_1(\mathbb{R}^n, E)\). A sufficient condition for \(b(\cdot)\) in order that \(b(D)\) generates an analytic semigroup in some Besov spaces is also given together with a number of applications to degenerate boundary value problems, operator convolution equations on the line, maximal regularity for parabolic Cauchy problems and so on.