Short answer is no.
L O N G answer: It depends on how deep you want to get into the music, level of playing / desired level of playing, how playing only the bass is working out for you, how good your brain is at understanding music, how involved you'd like to be in the music writing or band arranging process, etc etc etc.
If you want to learn anything, I would start with singing (even if you plan to never do it in public). There are a lot of tone-deaf musicians out there. I'd recommend learning singing #1 to every instrumentalist (drummers included). Learn to sing the melody, your favorite bass lines, study and sing other instrument's lines. Try writing bass lines in your head, then sing it, then transcribe it to the bass. Sing scales, intervals, harmonies, and take your time to really make sure you are hearing the note in your head and singing it in pitch.
Upright bass is a great one! It will easy to find the notes as it's tuned the same. Its a different animal to play, but having skills on one will give you great insight on the other, and vice-verca. Basslines don't always translate exactly between the instruments, so learn proper technique from a teacher.
Guitar is a natural choice for many bass players, and it will help you learn your relationship with it as a bassist. No need to learn to shred, but basic chord stuff is good to know; with in a song you should at least be know the roots of the chords and better what the chord value is (major, minor, etc), and better still to incorporate it with singing a melody and carrying a tune by yourself...not always practical at the campfire with your electric bass.
Mandolin is basically a 4 string instrument; 8 strings, in unison pairs of 2. Many bass players find it easy once you learn a few songs.
Piano is a universal instrument. Anyone studying music at a college level is required to learn piano in addition to their primary instrument.
Drums. Study drummers. Most drummers love to talk about gear, ask them how their stuff works, how they tune their drums or get cool tones.
Of course its all optional. I can dabble in all of the above, and although I wouldn't want to be seen doing most of it in public, at least I can transcribe music by ear, make arrangements, correct other players bad notes or help them find the best thing to play, participate in backup vocals, and make money playing and teaching multiple instruments, plus I just feel like I can relate to everyone else in the group and play the best possible bass lines, knowing what everyone else is doing.
