When using an array of dummy variables in regression analysis, do I need to drop one of the dummies if there are missing values in the data?

Question

I'm running a fixed effects regression with my dependent variable as life satisfaction score and main independent variable as education level. The education variables are broken down into the following categories:

• Degree
• Other higher degree
• A Level etc.
• GCSE etc.
• Other qualification
• No qualification
• Inapplicable
• No response

I am only including dummy variables for degree down to no qualification, with the assumption the those with inapplicable or no response answers are captured in the constant/intercept. Am I correct in thinking that this means that I will not fall into the dummy variable trap?