Short answer. Generally yes, but with limits. Rule 132, Section 3 gives a witness the right "not to give an answer which will tend to degrade his or her reputation, unless it be to the very fact at issue or to a fact from which the fact in issue would be presumed." A witness must still answer about a prior final conviction.
What the law says
Not to give an answer which will tend to degrade his or her reputation, unless it be to the very fact at issue or to a fact from which the fact in issue would be presumed
Rule 132, Section 3 — Rights and obligations of a witness. Read the full provision →
What the law says
But a witness must answer to the fact of his or her previous final conviction for an offense
Rule 132, Section 3 — Rights and obligations of a witness. Read the full provision →
The right this rule gives you
Rule 132, Section 3 lists this specifically among a witness's rights: "Not to give an answer which will tend to degrade his or her reputation, unless it be to the very fact at issue or to a fact from which the fact in issue would be presumed." As a general rule, you may decline to answer a question specifically because the answer would degrade your reputation.
Why this right has a built-in exception
The right is not unconditional — it stops at the words "unless it be to the very fact at issue or to a fact from which the fact in issue would be presumed." If the reputation-damaging answer would itself be the very fact the case turns on, or a fact from which that central fact would be presumed, the right to withhold it does not apply. Protecting your reputation cannot be used to keep the actual matter in dispute out of the record.
The other rights the same section lists
Section 3 groups the reputation right with several others: to be protected from irrelevant, improper, or insulting questions and from harsh or insulting demeanor; not to be detained longer than the interests of justice require; not to be examined except as to matters pertinent to the issue; and not to give an answer that would tend to subject the witness to a penalty for an offense, unless otherwise provided by law. All of these sit alongside the general duty that a witness must answer, even if the answer establishes a claim against them.
One thing you must answer regardless
The section closes with a specific carve-out that overrides even the reputation right: "But a witness must answer to the fact of his or her previous final conviction for an offense." Whatever damage that fact might do to your reputation, the right to withhold a degrading answer does not extend to your own prior final conviction — that specific fact has to be answered.