Achieving direct imaging of the annihilation position of a positron on an event-by-event basis using an ultrafast detector would have a great impact on the field of nuclear medicine. Cherenkov emission is the most attractive physical phenomenon for realizing such an ultrafast timing performance. Moreover, a microchannel-plate photomultiplier tube (MCP-PMT) is one of the most promising photodetectors for fully exploiting the fast timing properties of Cherenkov emission owing to its excellent single photon time resolution of 25 ps full width at half maximum (FWHM). However, as the MCP structure generally contains a lead compound, the gamma rays frequently and directly interact with the MCP, resulting in the degradation of its timing performance and generation of undesirable side peaks in its coincidence timing histogram. To overcome this problem, we have developed a new MCP-PMT based on an MCP consisting of borosilicate glass, thus drastically reducing the probability of the photoelectric effect occurring in the MCP. To evaluate its insensitivity to gamma rays and its timing performance, a coincidence experiment was performed and showed that the probability of direct interactions was reduced by a factor of 3.4. Moreover, a coincidence time resolution of 35.4 ± 0.4 ps FWHM, which is equivalent to a position resolution of 5.31 mm, was obtained without any pulse height/area cut, improving to 28.7 ± 3.0 ps when selecting on the highest amplitude events by careful optimization of the voltage divider circuit of the new MCP-PMT. The timing performance of this new MCP-PMT presents an important step toward making direct imaging possible.