We report measurements of the magnetic penetration depth λ in single crystals of CeCoIn5 down to ∼0.14 K using a tunnel-diode based, self-inductive technique at 28 MHz. While the in-plane penetration depth tends to follow a power law, λ // ∼ T 3/2 , the data are better described as a crossover between linear (T ≫ T * ) and quadratic (T ≪ T * ) behavior, with T * the crossover temperature in the strong-coupling limit. The c-axis penetration depth λ ⊥ is linear in T, providing evidence that CeCoIn5 is a d-wave superconductor with line nodes along the c-axis. The different temperature dependences of λ // and λ ⊥ rule out impurity effects as the source of T * .The compounds CeMIn 5 (M = Co, Ir, Rh) have recently been added to the heavy-fermion family, and have attracted much interest due to their similarity with the cuprates: quasi-2D structure and proximity to magnetic order [1]. CeCoIn 5 , in particular, is a good candidate for study: its superconductivity is not sensitive to small changes in unit-cell volume or composition, unlike CeCu 2 Si 2 , and it has the highest T c (∼2. Recently, there has been mounting evidence for unconventional superconductivity in CeMIn 5 . Specific heat data reveal a T 2 term at low temperature, consistent with the presence of line nodes in the superconducting energy gap [3]. Thermal conductivity measurements with in-plane applied field show four-fold symmetry, consistent with nodes along the (±π, ±π) positions [4]. NQR measurements show that there is no Hebel-Slichter peak just below T c [5]. Below T c the spin susceptibility is suppressed, indicating singlet pairing [5,6]. However, there are some ambiguities in some of the measurements. Thermal conductivity data yield a T 3.37 low-temperature behavior, that the authors claim is close to T 3 behavior predicted for unconventional superconductors with line nodes in the clean limit [3]. NQR measurements did not show the T 3 low-temperature behavior of 1/T 1 that is expected for a line node gap; instead 1/T 1 saturates below 0.3 K [5]. Microwave measurements down to ∼0.2 K showed a non-exponential behavior, and the authors claimed that λ(T) ∼T below 0.8 K [7], though the data clearly show some curvature in that temperature range. Further, the field was applied along the abplane, so the shielding currents have both in-plane and inter-plane components. In this paper, we present highprecision measurements of in-plane λ // and inter-plane λ ⊥ penetration depths of CeCoIn 5 at temperatures down to 0.14 K. We find that λ // is best treated as a crossover from ∼T to ∼T 2 at a temperature T * . Combined with the result that λ ⊥ ∝ T , this gives strong evidence for non-local behavior in a d-wave superconductor as predicted by Kostzin and Leggett [8].Details of sample growth and characterization are described in Refs. [1,9]. Measurements were performed utilizing a 28 MHz tunnel diode oscillator [10] with a noise level of 1 part in 10 9 and low drift. The magnitude of the ac field was estimated to be less than 5 mOe. The cryostat was surrounded by a...