The zero-temperature value of the in-plane London penetration depth, λab(0), has been measured in single crystals of Ba(Fe1−xCox)2As2 as a function of the Co concentration, x, across both the underdoped and overdoped superconducting regions of the phase diagram. For x≳0.047, λab(0) has been found to have values between 120±50 and 300±50 nm. A pronounced increase in λab(0), to a value as high as 950±50 nm, has been observed for x≲0.047, corresponding to the region of the phase diagram where the itinerant antiferromagnetic and superconducting phases coexist and compete. Direct determination of the doping-dependent λab(0) has allowed us to track the evolution of the temperature-dependent superfluid density, from which we infer the development of a pronounced superconducting gap anisotropy at the edges of the superconducting dome.