In this paper, we present and analyse optical photometry and spectra of the extremely luminous and slowly evolving Type Ia supernova (SN Ia) 2009dc, and offer evidence that it is a super-Chandrasekhar mass (SC) SN Ia and thus has a SC white dwarf (WD) progenitor. Optical spectra of SN 2007if, a similar object, are also shown. SN 2009dc had one of the most slowly evolving light curves ever observed for a SN Ia, with a rise time of ∼23 d and Δm15(B) = 0.72 mag. We calculate a lower limit to the peak bolometric luminosity of ∼2.4 × 1043 erg s−1, though the actual value is likely almost 40 per cent larger. Optical spectra of SN 2009dc and SN 2007if obtained near maximum brightness exhibit strong C ii features (indicative of a significant amount of unburned material), and the post-maximum spectra are dominated by iron-group elements (IGEs). All of our spectra of SN 2009dc and SN 2007if also show low expansion velocities. However, we see no strong evidence in SN 2009dc for a velocity ‘plateau’ near maximum light like the one seen in SN 2007if. The high luminosity and low expansion velocities of SN 2009dc lead us to derive a possible WD progenitor mass of more than 2 M⊙ and a 56Ni mass of about 1.4–1.7 M⊙. We propose that the host galaxy of SN 2009dc underwent a gravitational interaction with a neighbouring galaxy in the relatively recent past. This may have led to a sudden burst of star formation which could have produced the SC WD progenitor of SN 2009dc and likely turned the neighbouring galaxy into a ‘post-starburst galaxy’. No published model seems to match the extreme values observed in SN 2009dc, but simulations do show that such massive progenitors can exist (likely as a result of the merger of two WDs) and can possibly explode as SC SNe Ia.