We develop a string-net construction of a modular functor whose algebraic input is a pivotal bicategory; this extends the standard construction based on a spherical fusion category. An essential ingredient in our construction is a graphical calculus for pivotal bicategories, which we express in terms of a category of colored corollas. The globalization of this calculus to oriented surfaces yields the bicategorical string-net spaces as colimits. We show that every rigid separable Frobenius functor between strictly pivotal bicategories induces linear maps between the corresponding bicategorical stringnet spaces that are compatible with the mapping class group actions and with sewing. Our results are inspired by and have applications to the description of correlators in two-dimensional conformal field theories.