The desert certainly doesn't stop the suckers, speaking as a New Mexico resident. Ticks abound, Lyme equally so.
> Each dot represents one case of Lyme disease and is placed randomly in the patient’s county of residence. The presence of a dot in a state does not necessarily mean that Lyme disease was acquired in that state. People travel between states, and the place of residence is sometimes different from the place where the patient became infected.
https://www.cdc.gov/lyme/datasurveillance/tables-recent.html