The World Values Survey (WVS) is not a traditional panel—though it can be treated as such. It all depends upon how you want to aggregate your data. In each wave, their scientists interview roughly 1,200 people in each country. Each 'country' sample is representative of people 18 years-of-age and older residing within private households, regardless of their nationality, citizenship, or language. Thus, each wave of the WVS is a completely new sample of individuals from their respective country, and is more aptly considered repeated cross-sections data.
Could I make a fixed effects model by adding a dummy variable for "country", even though my data is not panel data?
Of course. Simply pool all waves together and estimate dummies for each country. Your software is indifferent to the structure of your data. It all depends on what variation you want to exploit. In many panel data contexts the first dimension i is often person and the second dimension t is often time. But it doesn't have to work this way. In fact, i could index over students/teachers and j could index over classrooms/universities. The second dimension doesn't even require intrinsic ordering as noted in a response here.
At the tail-end of your question you indicated that you do not have panel data. But this is not entirely accurate. For instance, you can create a 'pseudo-panel' by aggregating the individual data up to the country-level. If you do this by country and order it by survey wave, you technically have a "panel" dataset at the level of the country.