Is it possible for sciatica to cause a dragging foot?

Foot drop: If the nerves which cause the muscles of the foot to dorsiflex are irritated as part of the sciatica then a footdrop can develop. Typically if the L4 or L5 nerve roots as they exit the spline are irritated then pain will radiate into the leg and often times foot weakness will develop. Since there are many nerves which exit spine that can cause sciatica footdrop is uncommon with sciatica.

Yes: If this dragging of foot does not resolve, see your doctor right away as this could become permanent if continues for longer than a few days.