which oil for tire pumps?

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1,050
Location
Calif.
I've always been using a bicycle tire pump just about anything. From bicycle tires to car tires. Its probably more than 10 years old so and its getting sort of sticky as I pump it. So what kind of oil is okay to use to lube it? I was thinking maybe WD-40 or something. I'm assuming that some of the oil will get into the tires as it travels with the air. Any suggestions welcome Thanks.
 
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