Yes. An EU trademark registered with the EUIPO provides protection in all 27 EU member states simultaneously. If a third party infringes your mark in any member state, your EU trademark registration gives you legal grounds to act against them in that country.
EU trademarks are unitary: they cover all member states as a single unit. If the EUIPO refuses the application or if a national IP office raises a successful opposition based on a local earlier right, the EU trademark cannot be registered only in some member states and refused in others; it either succeeds or fails as a whole. In cases where a refusal is limited to specific member states, the application can be converted into national applications for the countries where it would have been accepted.