We report the detection of two elongated X-ray features, G359.89-0.08 and G359.54+0.18, in the Galactic center (GC) region using the Chandra X-Ray Observatory. G359.89-0.08 is an elongated X-ray feature located ~2' in projection south of the center of the Galaxy, Sgr A*. This X-ray feature source is partially coincident with a slightly curved ("wisplike") nonthermal radio source. The X-ray spectrum of G359.89-0.08 can best be characterized as nonthermal, with a photon index of 2. The morphological and spectral characteristics of the X-ray and radio emission associated with G359.89-0.08 are best interpreted as the synchrotron emission from a ram pressure–confined pulsar wind nebula. G359.54+0.18 is one of the most prominent radio nonthermal filaments (NTFs) in the GC region, located ~30' in projection from Sgr A*. A narrow (~10'') filament of X-ray emission appears to arise from one of the two strands that make up the radio NTF. Although the photon statistics are poor for this source, the X-ray emission is also likely to be nonthermal in nature. Several models for the production of X-ray emission in G359.54+0.18 are discussed.