The American Cancer Society has a long list of known and probable human carcinogenslist of known and probable human carcinogens. Among that long list are the following naturally occurring agents (exposure methods are also on the linked site):
- Beryllium
- Cadmium
- Chromium
- Epstein-Barr virus
- Ethanol in alcoholic beverages
- Helicobacter pylori (infection with)
- Hepatitis B virus (chronic infection with)
- Hepatitis C virus (chronic infection with)
- Human papilloma virus (HPV) types 16, 18, 31, 33, 35, 39, 45, 51, 52, 56, 58, 59 and 66 (Note: The HPV types that have been classified as carcinogenic to humans can differ by an order of magnitude in risk for cervical cancer)
- Human T-cell lymphotropic virus type I (HTLV-1)
- Neutrons
- Radon
- Schistosoma haematobium (flatworm; infection with)
- Solar radiation
- Aflatoxins (naturally occurring mixtures of)
- Alcoholic beverages
- Areca nut, Betel nut
- Coal-tar pitches
- Coal-tars
- Mineral oils, untreated and mildly treated
- Plants containing aristolochic acid
- Shale-oils
- Soots
- Tobacco, smokeless
- Wood dust
This is only a partial list of the naturally occurring agents and a smaller subset of all known carcinogens.
It can be readily shown that cancers can be caused by natural processes. @Konrad Rudolph has already described that cancer is strongly associated with age--the longer you live, the more likely you are to be exposed to a carcinogen.
Now the caveat: Although there are many naturally occurring carcinogens, there are many more additional carcinogens that exist in sufficient quantities to cause cancer only because of human industrial development. While it is incorrect to say that cancer is a man-made disease, you could claim that human industry has increased the risk-of-exposure to carcinogens, thus advancing the spread, or prevalence, of cancer. To be fair, such a claim should come with evidence showing that the particular carcinogen has caused cancer and that the carcinogen could not exist unless humans created or refined it.
[The last paragraph was edited to hopefully clear up misinterpretations of the more terse version that was originally submitted.]