Can plants get "cancer"?

They are made out of cells and exposed to UV rays nearly all the time. So why shouldn't the UV rays ionize their cell nucleus and corrupt the DNA?

Any complex multicellular organism can be suseptible to defects in cell proliferation causing individual cells or groups of cells to develop abnormally. In animals, these defects lead to tumour formation and cancer and plants develop tumours too which can be detrimental to how they function or develop. One crucial difference between plant and animal tumours is that, unlike many animal cells, plant cells are incapable of moving as they are fixed in a cell wall matrix. As such, the 'cancer' in plants is not able to spread to other parts of the organism and rarely kills.

So why do plants develops tumours? You asked in your question whether exposure to UV light could be a cause, but as far as I can find this is not the case. A vast majority of tumours developed by plants are caused by pathogens - specialised viruses and bacteria that invade plant cells and cause defects. One of the most common plant tumours are called Crown Galls which develop in the plant stems and are caused by a specific soil bacteria. Similarly, certain fungal diseases can also lead to the development of tumour like growths.

Certain plant varieties are also suspect to spontaneous tumour formation due to genetic disorders, particularly hybrid plants. Tobacco plants, for example, are one of the most suseptible to this and hybrid can be so over-run with tumours that flower and seed development is severely comporised.