PSR B1259−63 is a middle-aged radio pulsar (P = 48 ms, τ = 330 kyr, andĖ = 8.3 × 10 35 erg s −1 ) in an eccentric binary (P orb = 3.4 yr, e = 0.87) with a high-mass Be companion, SS 2883. We observed the binary near apastron with the Chandra ACIS detector on 2009 May 14 for 28 ks. In addition to the previously studied point-like source at the pulsar's position, we detected extended emission on the south-southwest side of this source. The point-like source spectrum can be described by the absorbed power-law model with the hydrogen column density N H = (2.5 ± 0.6) × 10 21 cm −2 , photon index Γ = 1.6 ± 0.1, and luminosity L 0.5-8 keV ≈ 1.3 × 10 33 d 2 3 erg s −1 , where d 3 is the distance scaled to 3 kpc. This emission likely includes an unresolved part of the pulsar wind nebula (PWN) created by the colliding winds from the pulsar and the Be companion, and a contribution from the pulsar magnetosphere. The extended emission apparently consists of two components. The highly significant compact component looks like a southward extension of the point-like source image, seen up to ∼4 from the pulsar position. Its spectrum has about the same slope as the point-like source spectrum, while its luminosity is a factor of 10 lower. We also detected an elongated feature extended ∼15 southwest of the pulsar, but the significance of this detection is marginal. We tentatively interpret the resolved compact PWN component as a shocked pulsar wind blown out of the binary by the wind of the Be component, while the elongated component could be a pulsar jet.