Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Can we find two positive constants $C$ and $\epsilon$, depending only on $n$, such that under the normal coordinates $(g_{ij})$ with respect to any point $p \in M$, we have $$ |\partial_k g_{ij}(x)| \le C $$ for any $|x| \le \epsilon$?

Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Can we find two positive constants $C$ and $\epsilon$, depending only on $n$, such that under the normal coordinates $(g_{ij})$ with respect to any point $p \in M$, we have $$ |\partial_k g_{ij}(x)| \le C $$ for any $|x| \le \epsilon$?

As pointed out in the comment, if the injectivity radius at $p$ is small, then the estimates should be understood for the pull-back of $g$ to the tangent space, which is always well-defined by the curvature bound.