We study the Yang--Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko--Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian loop in the group of unitary matrices.