No, you can get sub-pixel accuracy with a static image. The point is the star takes up multiple pixels, and when it moves by less than a pixel, some pixels get brighter while others get dimmer.

This is a common fallacy in computer vision. Often people get to a within a pixel and stop because they assume they can't get better. Most times you can get substantially better. I worked on a qr-code like system where the scanner could reconstruct the code from an image with (slightly) less pixels than 'pixels' in the code.

GP is right. The point spread function will spread the flux of a point source (eg. a star) across multiple pixels. You can identify the location of the actual star with sub pixel accuracy if you have a good PSF model.