The following strategy worked:
The following source code worked for getting a database file created:
ZConnection1.LibraryLocation := TPath.Combine(TPath.GetHomePath, 'libsqliteX.so');
DbLocation := TPath.Combine(TPath.GetDocumentsPath, 'db_sqlite.db');
Memo1.Lines.Append(DbLocation);
ZConnection1.Database := DbLocation;
ZConnection1.Protocol := 'sqlite';
ZConnection1.ClientCodepage := 'UTF-8';
ZConnection1.Connected := true;
Memo1.Lines.Append(BoolToStr(FileExists(DbLocation), True));
In the deployment settings tell Delphi to deploy the libsqliteX.so file to ".\assets\internal\".