Having spent several years in a small town, and currently working in a fairly small community, I can somewhat relate to your concern. However, it really would be best to have your current job on your resume.
As mhoran_psprep's comment says, reviewers will probably conclude that you have been unemployed since the job before your current one if you leave your current position off your resume. If you still get an interview and then tell your potential employer that you're really working, the interviewer may feel misled and stop considering you for the job since they will likely believe they cannot trust you.
Additionally, people looking to fill a position are generally most interested in your most recent experience. That is the freshest thing in your brain and probably what you're best at now. Thus, they will be looking to fit that experience against the job they need to fill.
While I am sure there are exceptions, people looking to fill jobs are usually discrete about resumes they receive. At least one reason is that they would not want it known that they are divulging the resume owners' names. This is because people will not want to risk submitting resumes to them if the submitters' names were getting out.
You might want to see this JobStar resume guide. I admit I'm not familiar with it, but it is the top recommended resume site of the author of What Color is your Parachute, which has been one of the top job hunting books for decades.