We perform a Bayesian analysis of pulsar-timing residuals from the NANOGrav pulsar-timing array to search for a specific form of stochastic narrow-band signal produced by oscillating gravitational potential (Gravitational Potential Background) in the Galactic halo. Such oscillations arise in models of warm dark matter composed of an ultralight massive scalar field (m ≃ 10−23 eV). The propagation of an electromagnetic signal from a pulsar through the time-dependent spacetime will leave an imprint in the pulsar timing, much like a gravitational wave. From the physical point of view, this is the classical Sachs-Wolfe effect. A distinctive feature of the pulsar-timing residuals due to GBP produced by a variable scalar field is that the amplitude of the TOA residuals should be independent of the pulsar location in the sky. In the monochromatic approximation, the stringent upper limit (95% C.L.) on the variable gravitational potential amplitude is found to be (Ψc<1.14 × 10−15), corresponding to the characteristic strain hc = 2$\sqrt{3}$Ψc < 4 × 10−15 at f=1.75 × 10−8 Hz. In the narrow-band approximation, the upper limit of this background energy density is ΩGPB < 1.27 × 10−9 at f=1.75 × 10−8 Hz. These limits are an order of magnitude higher than the expected signal amplitude assuming all Galactic dark matter is made of such scalar particles. The applied analysis of the pulsar-timing residuals can be used to search for any narrow-band stochastic signals with different correlation properties. As a by-product, parameters of the red noise present in four NANOGrav pulsars were found.