  PSR J1809-1917 is a young ($\tau=51$ kyr) energetic
($\dot{E}=1.8\times10^{36}$ erg s$^{-1}$) radio pulsar powering a pulsar wind
nebula (PWN). We report on the results of three Chandra X-ray Observatory
observations which show that the PWN consists of a small ($\sim 20''$) bright
compact nebula (CN) and faint extended emission seen up to $2'$ from the
pulsar. The CN is elongated in the northeast-southwest direction and exhibits
morphological and flux variability on a timescale of a few months. We also find
evidence of small arcsecond-scale jets extending from the pulsar along the same
direction, and exhibiting a hard power-law (PL) spectrum with photon index
$\Gamma_{\rm jet}=1.2\pm0.1$. The more extended emission and CN share the same
symmetry axis, which is also aligned with the direction toward the TeV
$\gamma$-ray source HESS J1809--193, supporting their association. The spectrum
of the extended nebula (EN) fits an absorbed PL with about the same slope as
that of the CN, $\Gamma_{\rm CN}\approx\Gamma_{\rm EN}=1.55\pm0.09$; no
spectral changes across the EN's 2 pc extent are seen. The total PWN 0.5-8 keV
luminosity is $L_{\rm PWN}\approx 9\times10^{32}$ erg s$^{-1}$, about half of
which is due to the EN.
