I don't think any common camera gives you that kind of control. Mine certainly doesn't.

The basic problem is that a strong bias level is added to black. Digital sensors are basically linear, so even if the picture is exposed so that the strongest highlight it just at the top end of the sensor's range, the bottom end of the range below the black offset won't be used.

I just analyzed your picture, and the black level is .363, .376, .406. As proof, here is your picture with that value used as the black level:

That wastes at most .4 of your sensor range, which is less than one bit of resolution.

The white level in your original is .852, .856, .910. That represents a small but reasonable unused sensor range at the high end. Here is your picture with both black and white expanded to the ends of the range:

Note that considering both the black and white ends of the range, you lost about 1 bit of sensor resolution. If you have a 14 bit sensor, for example, then this picture actually has 13 bits of brightness resolution. For a 12 bits sensor, you ended up with 11 bits.

Even just 11 bits is still plenty since most ways of displaying the image won't be good to any more than 8 bits or so. That leaves a factor of 8 (3 bits) resolution overhead for things like non-linear brightness adjustments and other effects you might want to add. The point is that loosing 1 bit from the maximum sensor range is perfectly fine and normal, is the kind of thing you should expect, and in the end isn't hurting your picture. You would end up with the same captured detail if the darkest parts of the picture were really black but you underexposed it by 1 f-stop. Again, 1 bit of dynamic range loss should be well within expectations and what you can deal with in your normal workflow.