The capability of plasmas to sustain ultrahigh electric fields has attracted considerable interest over the last decades and has given rise to laser-plasma engineering. Today, plasmas are commonly used for accelerating and collimating relativistic electrons, or to manipulate intense laser pulses. Here we propose an ultracompact plasma undulator that combines plasma technology and nanoengineering. When coupled with a laser-plasma accelerator, this undulator constitutes a millimetre-sized synchrotron radiation source of X-rays. The undulator consists of an array of nanowires, which are ionized by the laser pulse exiting from the accelerator. The strong charge-separation field, arising around the wires, efficiently wiggles the laser-accelerated electrons. We demonstrate that this system can produce bright, collimated and tunable beams of photons with 10-100 keV energies. This concept opens a path towards a new generation of compact synchrotron sources based on nanostructured plasmas.