PSR B1259-63 is a middle-aged radio pulsar (P=48 ms, tau=330 kyr, Edot=8.3*10^{35} erg/s) 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 pointlike source at the pulsars position, we detected extended emission on the south-southwest side of this source. The pointlike 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 Gamma = 1.6+/-0.1, and luminosity L_{0.5-8 keV} = 1.3*10^{33} d_3^2 erg/s, 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 pointlike source image, seen up to about 4 arcsec from the pulsar position. Its spectrum has about the same slope as the pointlike source spectrum, while its luminosity is a factor of 10 lower. We also detected an elongated feature extended ~15 arcsec southwest of the pulsar, but 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.
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