I think it's important to keep two things separate here:  

First, if $(M,F)$ is a smooth Finsler manifold (which means that $F^2:TM\to [0,\infty)$ is smooth and strongly convex away from the zero section of $TM$), then the unit sphere bundle $\Sigma\subset TM$ (aka the *tangent indicatrix*) is a smooth hypersurface in $TM$ and there is an open neighborhood $U$ of $\Sigma\times\{0\}$ in $\Sigma\times\mathbb{R}$ on which there is defined a smooth mapping $\exp:U\to M$ such that, for each fixed $u\in \Sigma$, the curve $\gamma_u(t) = \exp(u,t)$ (defined for all $t\in\mathbb{R}$ such that $(u,t)$ lies in $U$) is the maximally extended unit speed Finsler geodesic with initial velocity $u\in\Sigma$.

Second, for any given $p\in M$, there is a $\delta>0$ such that, if $B_\delta(p)\subset T_pM$ is the set of vectors $v\in T_pM$ satisfying $F(v)<\delta$, then there is a well-defined mapping $\exp_p:B_\delta(p)\to M$ such that $\exp_p(0_p) = p$ and $\exp_p(tu) = \exp(u,t)$ for all $u\in \Sigma_p$ and all $t\in(0,\delta)$.  By taking $\delta$ sufficiently small, one can ensure that $\exp_p:B_\delta(p)\to M$ is a homeomorphism onto its (open) image, one that is a smooth diffeomorphism away from $0_p$.  

However, unless $F$ is reversible (i.e., $F(-v) = F(v)$), the map $\exp_p:B_\delta(p)\to M$ need not be $C^2$ at $0_p$, because it can easily happen that $\exp(-u,-t)\not=\exp(u,t)$, so the $\exp_p$-image of the line segment $\{tu\ |\ |t|<\delta\}$ need not be a $C^2$ curve in $M$ (it will be $C^1$, though).

Coming up with a simple, explicit example, however, is not easy because, in general, it is not easy to integrate the geodesic equations and compute the map $\exp:U\to M$ explicitly.  In the one case where it is easy, namely the case of a Minkowski Finsler metric, i.e., $M=\mathbb{R}^n$ and $F$ is translation invariant, the exponential map is just $\exp(p,(u,t)) = p+tu$ for $p\in \mathbb{R}^n$ and $u\in \Sigma_p\subset T_p\mathbb{R}^n = \mathbb{R}^n$, so $\exp_p$ is a smooth diffeomorphism in this case, regardless of how non-Riemannian $F$ is.  (This shows that it's not just a matter of looking at the shape of the unit sphere at one point of $M$.)

However, in an answer to [this question][1], I gave an example of a Finsler metric $F$ on a surface $M$ that is reversible and has the property that $F^4$ is a smooth function on $TM$, but the fourth power of the Finsler distance function from a given point of $M$ is *not* smooth.  Thus, this is an example where the map $\exp_p$ is not smooth at the origin $0_p$. 


  [1]: http://mathoverflow.net/questions/220887/smoothness-of-the-fourth-power-of-the-geodesic-distance-in-a-finsler-geometry/224704#224704