Not all do.
The Canon¹ EOS 1D X and EOS 1D X Mark II, for instance, have limited options that allow for shooting bursts at their maximum frame rate only with the mirror locked up.
- The EOS 1D X has a maximum frame rate of 12 fps using the viewfinder with the mirror flipping between each frame. It can do 14 fps with the mirror remaining locked up between each frame during viewfinder shooting.
- The EOS 1D X Mark II has a maximum frame rate of 14 fps using the viewfinder with the mirror flipping between each frame. It can do 16 fps with the mirror remaining locked up between each frame during viewfinder shooting.
Even at longer exposure times, when the length of time the shutter remains open reduces the maximum number of fps², if the camera is set to shoot at the maximum "locked up" frame rate, the mirror will not cycle between frames.
Of course there is no autofocus or tracking done between frames: the exposure and AF position is set before the first image is taken and stays the same for the entire series.
There are many other DSLRs that can take multiple images without flipping the mirror down. The camera just needs to be in "Live View" mode with any autofocusing mode that flips the mirror down to use the dedicated PDAF system turned off.
Most Canon¹ DSLRs with Live view capability include this option of using imaging sensor based contrast detection AF or Dual Pixel CMOS AF on some models.
¹ I use examples from Canon's lineup because it is what I am most familiar with. The makers of other DSLRs offer similar options with at least some of their models.
² For example, if an exposure time of 1/4 second is selected the camera will be limited to less than 4 fps because it takes one second, plus the shutter cycle time, to do four frames at 1/4 second each. However, if 16 fps is selected in the maximum frame rate menu option for the 1D X Mark II, the mirror will not cycle between each 1/4 second exposure.