  We construct a tangent bundle exponential map and locally autoparallel
coordinates for geometries based on a general connection on the tangent bundle
of a manifold. As concrete application we use these new coordinates for
Finslerian geometries and obtain Finslerian geodesic coordinates. They
generalise normal coordinates known from metric geometry to Finsler geometric
manifolds and it turns out that they are identical to the Douglas-Thomas normal
coordinates introduced earlier. We expand the fundamental geometry function of
a Finsler spacetime in these new coordinates and find that it is constant to
quadratic order. The quadratic order term comes with the non-linear curvature
of the manifold. From physics these coordinates may be interpretation as the
realisation of an Einstein elevator in Finslerian spacetime geometries.
