Lamp-pumped rubidium atomic frequency standard (RAFS) is one of the most commonly utilized atomic frequency standards. Over the past few decades, the RAFS's frequency stability performance has improved rapidly, and the best one has been in the 10 -13 τ -1/2 level. In this article, we demonstrate a RAFS with stability in the 10 -14 τ -1/2 level for the first time. In design of the physics package, a rubidium spectral lamp with Xe as the starter gas was used as the pumping light source. The light was filtered by using optical and isotope double-filtering technique. A large slotted tube microwave cavity and a rubidium absorption cell with a diameter of 40 mm were utilized to enhance the atomic discrimination signal. A sealed box was designed for the physics package to isolate it from the barometric environment. A low phase noise 6.834xx GHz microwave was employed to interrogate the rubidium clock transition. Based on quantitative analysis of the signal-to-noise ratio of the atomic discrimination signal and the phase noise of the interrogation microwave, the stability of the RAFS was predicted to be 7.6×10 -14 τ -1/2 . The short-term stability of the RAFS was measured by using a hydrogen maser and an optical microwave generator as references, results are 9.0×10 -14 τ -1/2 (1 ~ 100 s) and 9.1×10 -14 τ -1/2 (1 ~ 100 s) respectively. The measured results are in agreement with the predicted one.