Is speed of light not constant in a strong gravitational field?

The physical speed of light in vacuum c is always constant and does not change, even in the presence of a gravitational field. The same holds for the vacuum magnetic permeability and for the electric permittivity.

Some confusion may come from the curvature of space and curvilinear coordinates. In presence of gravity, the spacetime is curved and one needs curvilinear coordinates to describe motion in it. The light always propagates along null-geodesics which are known to be coordinate independent. The coordinate speed $dx/dt$, however, can be arbitrary (since in General Ralativity one can work with any arbitrary coordinates). But the speed of light measured locally by the inertial observers (called also free-falling observers, feeling no forces) will be always the same.