= Solution
A <local isometry> here is a local diffeomorphism whose pullback metric is the source metric. Use the usual convention that a <covering map> is surjective. Assume the target is connected and the source is nonempty; this is the connected-manifold setting of the intended assertion. For a disconnected target the precise result is a covering of the union of target components met by the map. For example, inclusion $\mathbb R\hookrightarrow\mathbb R\sqcup\mathbb R$ is a <local isometry> from a complete manifold but is not a surjective covering of the whole target.
The correct <Hopf-Rinow theorem> says that on a connected finite-dimensional <Riemannian manifold>, completeness for the induced distance is equivalent to <geodesic completeness>, and to compactness of every closed bounded subset. These conditions imply that every two points are joined by a <minimizing geodesic>. The last property by itself does not imply completeness: an open Euclidean ball has minimizing segments between every pair but is incomplete.
A <local isometry> preserves the <Levi-Civita connection>, by uniqueness of the torsion-free metric connection on each <local isometry> chart. Hence it preserves affinely parametrized <geodesics>, and
$$
f(\exp_x v)=\exp_{f(x)}(df_xv)
$$
whenever the source <geodesic> is defined. This is <geodesic preservation by a local isometry>.
Suppose $M$ is complete. Given $y\in N$, join $f(x)$ to $y$ by a minimizing target <geodesic> using target completeness. The source <geodesic> whose initial velocity is its inverse image under $df_x$ exists for its full parameter interval by source completeness; its image is the target <geodesic> by uniqueness. Thus $f$ is onto.
Choose a normal ball $U=\exp_y(B_r(0))$ with $\exp_y$ a diffeomorphism on $B_r(0)\subset T_yN$. For every $x\in f^{-1}(y)$ put $U_x=\exp_x(B_r(0))$. Source completeness defines these exponential maps throughout the tangent balls. The displayed exponential identity makes $f:U_x\to U$ a diffeomorphism: injectivity follows from injectivity of the target <exponential map>, and differentiating the identity shows the source exponential has invertible derivative on the ball, so its image is open.
These sheets are disjoint. If $z$ lay in two, lift the reversed radial target <geodesic> from $f(z)$ to $y$, starting at $z$. Its initial velocity is uniquely fixed by $df_z$; uniqueness of the source <geodesic> forces the two sheet centres to be equal. They exhaust $f^{-1}(U)$: for any point over $U$, lift that reversed radial <geodesic> using source completeness, obtaining a centre over $y$, and then reverse it. Thus $U$ is evenly covered. This proves the <complete local isometry is a covering> assertion.
Conversely suppose $f$ is a covering. Given any initial tangent vector in $M$, the corresponding target <geodesic> extends over all $\mathbb R$ by target completeness. The <path lifting theorem> for coverings lifts it over every compact time interval, with uniqueness gluing the lifts to a curve on $\mathbb R$. In local covering charts the lift is a <geodesic>, because the charts are <local isometries>. It extends the source initial-value <geodesic>. Therefore $M$ is geodesically complete, and Hopf-Rinow yields metric completeness on each source component. In the stated connected-target setting,
$$
\boxed{M\text{ is complete}\iff f\text{ is a covering map}.}
$$
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