Botox Affecting Upper Lip Movement?

If this is from the Botox inection don’t panic. It will go away, but in can take a few months. If the Crows feet are injected at the bottom and your muscle which allows you to smile is a strong muscle and is prominent near the lower Crows feet than it may have been temporarily affected.

Generally, when Botox is injected for treatment of the crow's feet, it is placed superficially into the orbicularis oculi muscle. Very deep to these alon the cheek bone are the zygomatic muscles which move the upper lip. Botox usually remains very localized and it is unlikely that it would spread this deep if injected superficially but it is a possiblity.
The difficult thing to predict about Botox is that no matter who injects it, it can spread up to 3 cm from the injection site. It is kind of similar to predicting an ink spot.
So the answer to your question is, yes it is possible but not likely in the hands of an experienced injector.

If the crow's feet are injected with Botox too far down the cheek, it can affect the muscles that raise the cheek when smiling.  The result is a drooping of the mouth on the affected side.  The effect will probably last 8 to 10 weeks and will then resolve.  Please see an experienced dermatologist or plastic surgeon for your treatments.