The thermal conductivity κ of the iron-arsenide superconductor Ba(Fe1−xCox)2As2 was measured down to 50 mK for a heat current parallel (κc) and perpendicular (κa) to the tetragonal c axis for seven Co concentrations from underdoped to overdoped regions of the phase diagram (0.038≤x≤0.127). A residual linear term κc0/T is observed in the T→0 limit when the current is along the c axis, revealing the presence of nodes in the gap. Because the nodes appear as x moves away from the concentration of maximal Tc, they must be accidental, not imposed by symmetry, and are therefore compatible with an s± state, for example. The fact that the in-plane residual linear term κa0/T is negligible at all x implies that the nodes are located in regions of the Fermi surface that contribute strongly to c-axis conduction and very little to in-plane conduction. Application of a moderate magnetic field (e.g., Hc2/4) excites quasiparticles that conduct heat along the a axis just as well as the nodal quasiparticles conduct along the c axis. This shows that the gap must be very small (but nonzero) in regions of the Fermi surface which contribute significantly to in-plane conduction. These findings can be understood in terms of a strong k dependence of the gap Δ(k) which produces nodes on a Fermi-surface sheet with pronounced c-axis dispersion and deep minima on the remaining, quasi-two-dimensional sheets.