In the strict sense, no, because the same digital numbers could have been produced in many ways, even completely artificially (not from measuring light). Therefore, unlike the grooves on a bullet or fingerprints, deliberate deception is easier to do and harder to detect.
If you can rule out deliberate deception, then I think it could be possible, depending on the characteristics of the scene and exposure.
First, just the image size can tell you a lot. Especially for high end cameras, the exact number of pixels in X and Y greatly narrows it down, possibly even to a particular model.
Other possibilities include looking for a noise signature in low-light regions of the image. Of course this is difficult if there aren't any or they are small. This is assuming you have the raw file (with metadata stripped). Compression artifacts will likely swamp small noise signatures.
Again assuming you have the raw image, just the Bayer pattern layout will eliminate many camera models. The number of scan lines per block, assuming TIFF-style format, also varies between models. So do other somewhat arbitrarily-chosen characteristics that don't matter to the photographer. Even the order of TIFF tags and the offset of the data fields into the file are identifiying characteristics. There are a lot of these little things that probably go a long way to narrowing down the model of camera.
There is a good chance that the sensor had some dirt somewhere. Just one obviously-off pixel could be a pretty conclusive if you can show that the particular camera in question has dirt at that location. Take a blank picture with the test camera, then look for any anomalies. Most likely you'll find a few. Especially if they can be traced to dirt on the sensor, that should be as good as a fingerprint. The chance of two cameras of the same model having even just two dirt spots in exactly the same place is astronomically small.