Cold molecular gas has recently been detected in several cooling flow clusters of galaxies containing huge optical nebula. These optical filaments are tightly linked to cooling flows and related phenomena, such as rising bubbles of relativistic plasma fed by radio jets. We present here a map, in the CO(2-1) rotational line, of the cold molecular gas associated with some of the Hα filaments surrounding the central galaxy of the Perseus cluster: NGC 1275. The map, extending to about 50 kpc (135 arcsec) from the center of the galaxy, has been made with the 18-receiver array HERA at the focus of the IRAM 30 m telescope. Although most of the cold gas is concentrated to the center of the galaxy, the CO emission is also clearly associated with the extended filaments conspicuous in ionised gas, and could trace a possible reservoir fueling the star formation there. Some of the CO emission is also found where the X-ray gas could cool down more efficiently at the rims of the central X-ray cavities (where the hot gas is thought to have been pushed out and compressed by the expanding radio lobes of the central AGN). The CO global kinematics do not show any rotation in NGC 1275. The cold gas is probably a mixture of gas falling down on the central galaxy and of uplifted gas dragged out by a rising bubble in the intracluster medium. As recently suggested in other cluster cores, the cold gas peculiar morphology and kinematics argue for the picture of an intermittent cooling flow scenario where the central AGN plays an important role.