Plugging in the GPS coordinates from the source in psnz's answer, I ended up getting 135.7km or 84.3 miles (the same in two online distance calculators).
Taking this, I thought it would be interesting to see how the distance changes if instead of as the crow flies, we accounted for the slant the line would take it we connected the peak of Mt Whitney to the lowest point in Badwater Basin. A simple application of the Pythagorean theorem using 135.7km as the base and 4.506km as the vertical distance between the high and low points gives us 135.8km (to 4 significant figures). So an extra 100m if you wanted to actually travel the minimum distance between the two points (though I suspect that line probably intersects with the ground at some point so it wouldn't be physically possible for a mere human anyway - though something like a neutrino would have no trouble!).
I know usage varies, but at least some people (and the USGS) acknowledge that Alaska is on the same continent as the contiguous 48 states, in which case the highest point at Denali (Mount McKinley) is much further away than Mount Whitney from the lowest point in Death Valley. The distance between those points is calculated by https://www.calcmaps.com/map-distance/ as about 3779 km or 2348 mi.
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