Below is a sample of what I mean. The light source is behind the object and lower than the sign. How would I draw a cast shadow for the sign?
You have a little shaked things, but we can restore the order with a ray drawing. Let's assume your horizon is the theoretical perspective drawing horizon line and let's assume the sun is so distant that rays from it are parallel:
The top of the post is in higher altitude than observer's eye. The cyan line is horizontal. It points the theoretical horizon line on the image plane.
In the image the sun apparently has lower elevation than the top of the post. That means the green ray from the sun to observer's eye meets the image plane at lower than the image of the top of the post. But the ray comes to the eye from very high (=infinite) altitude because there's an angle between the ray and the horizontal sight line (=cyan). The commentators have already said that fact. Thus the shadow of the post cannot be infinitely long on the ground.
On the ground the shadow of the post reaches much further from the image plane than the station point. The shadow of the post in the image goes beyond the edge of the paper for that reason. I have marked point X on the post. The shadow of point X is at the bottom edge of the image. Only very little piece of the whole shadow that the post makes fits into the image.
To construct the shadow of the post to your image there's no need to draw a full perspective drawing from the assumed places and sizes of the things. We get something also without them:
We draw a vertical line from the sun to the horizon line and start rays for the edges of the shadow from the crossing to the bottom of the post. The blue lines are drawn to find point X. It's needed to see what's included to the visible part of the shadow.