I need to show that every smooth vector field along an immersion has local smooth extensions.
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If you do not demand nonvanishing, there is never a problem to extend a local section of a vector bundle to a global section. Use local trivializations to extend them locally and glue these together with a partition of unity. 


Consider such an immersed curve (without endpoints!) and its tangent vector field. This field cannot have smooth extension around the points where the curve is almost touching itself. 

