1.55mm (0.062") pins with 5mm spacing is what I was guessing. But as the last poster stated, get an overall length of the longest connector on the board to see if this is really 5mm or 0.2" spacing (measure center of pin on one end to center of pin on the opposite end of connector).
For 1.55mm pins with 5mm spacing - my 'oddball' connector source has a version that goes to 6 pins in length only. But these would not look at all like what you have now - see link Ken posted for a similar connector (10/5 mm spacing).
For 0.2" pin spacing - all I can find are square pins. Available up to 20 pins long (like Molex SPOX). Again, these won't look like what you have now - these would look more like standard 0.156" plugs used nearly everywhere.
The problem with both of these alternatives - they are both much thicker than what you are using now. I'm sure the silkscreens would be partially covered and hopefully no interference with nearby resistors. Can you get any pins out without mangling it or the connector? I have a source that makes hundreds of crimp contact varieties, maybe we can find a close match.