Yes, asthma can develop later in life.

More Info: Though asthma is far more prevalent in children than adults, adults can develop asthma even if they have displayed no signs or symptoms earlier in life. When asthma develops in anyone over the age of 20, it is referred to as adult-onset asthma. 

Adult-Onset Asthma   WebMD

What Is Adult Onset Asthma?

The term adult-onset asthma refers to an asthma diagnosis after the age of 20. Most adults who are diagnosed with adult-onset asthma also suffer from allergies, but it can also occur as a result of exposure to irritants at the workplace known as occupational asthma, or allergens present in the home environment. It is also possible for women who are pregnant or are menopausal to develop adult-onset asthma, which can occur as a result of hormonal changes within the body. Women who are on an estrogen therapy regimen may also develop adult-onset asthma, as will people who have recently had a cold or the flu.

Who Is More Prone to Adult-Onset Asthma?

Women who are going through hormonal changes, people who suffer from allergies or obesity and people with influenza are more prone to adult onset asthma. It is more common in women, especially those that are pregnant or going through menopause. Adult onset asthma also occurs in women that are taking estrogen supplements. Long periods of exposure to smoke, perfume, or other irritants can also trigger asthma. Some adults develop asthma due to anxiety or strenuous exercise. In order to be diagnosed with "adult" onset asthma an individual must be over 20 years old.